( 0,0 ) .,title="fig : " ]  # 1#2#3#4#5   ( 3676,2368)(0,-1483 )  in this work we are concerned with two topics whose name contains the word `` smooth '' , but in totally different meaning .the first is _ balancing ( smoothing ) networks _ , the second is _ smoothed analysis_. let us start by introducing these two topics , and then introduce our contribution interrelating the two .   in the standard abstraction of _ smoothing _ ( balancing ) networks  , processors are modeled as the vertices of a graph and connection between them as edges .each process has an initial collection of unit - size jobs ( which we call tokens ) .tokens are routed through the network by transmitting tokens along the edges according to some local rule .we measure the quality of such a balancing procedure by the maximum difference between the number of tokens at any two vertices at the end .the local scheme of communication we study is a _ balancer _gate : the number of tokens is split as evenly as possible between the communicating vertices with the excess token ( if such remains ) routed to the vertex towards which the balancer points .more formally , the balancing network consists of vertices , and matchings ( either perfect or not ) .we associate with every matching edge a balancer gate ( that is , we think of the edges as directed edges ) . at the beginning of the first iteration , tokens are placed in vertex , and at every iteration , the vertices of the network perform a balancing operation according to the matching ( that is , vertices and interact if ) .one motivation for considering smoothing networks comes from the server - client world .each token represents a client request for some service ; the service is provided by the servers residing at the vertices .routing tokens through the network must ensure that all servers receive approximately the same number of tokens , no matter how unbalanced the initial number of tokens is ( cf . ) . more generally ,smoothing networks are attractive for multiprocessor coordination and load balancing applications where low - contention is a requirement ; these include _ producers - consumers _ and distributed numerical computations . together with _ counting networks _ , smoothing networks have been studied quite extensively since introduced in the seminal paper of . initiated the study of the _  network _( cube - connected - cycles , see figure [ fig : ccc16 ] ) as a smoothing network - network and therefore we will stick to the latter in the following . ] .for the special case of the , sticking to previous conventions , we adopt a `` topographical '' view of the network , thus calling the vertices _ wires _ , and looking at the left - most side of the network as the `` input '' and the right - most as the `` output '' . in the , two wires at layer are connected by a balancer if the respective bit strings of the wires differ exactly in bit .the  is a canonical network in the sense that it has the smallest possible depth ( number of rounds ) of as a smaller depth can not ensure any discrepancy independent of the initial one .moreover , it has been used in more advanced constructions such as the _ periodic ( counting ) network _  .  as it turns out , the initial setting of the balancers directions is crucial .two popular options are an arbitrary orientation or a uniformly random one . a maximal discrepancy of was established for the for an arbitrary initial orientation  . for a random initial orientation of the , a discrepancy of at most for the ( this holds _ whp _ we mean with probability tending to as goes to infinity . ] over the random initialization ) , which was improved by to ( and a matching lower bound ) .results for more general networks have been derived in for arbitrary orientations . for expander graphs ,they show an -discrepancy after rounds when is the discrepancy of the initial load vector .this was recently strengthened assuming the orientations are set randomly and in addition the matchings themselves are chosen randomly .specifically , for expander graphs one can achieve within the same number of rounds a discrepancy of only .let us now turn to the second meaning of `` smoothed '' . smoothed analysis comes to bridge between the random instance , which typically has a very specific `` unrealistic '' structure , and the completely arbitrary instance , which in many cases reflects just the worst case scenario , and is thus over - pessimistic in general . in the smoothed analysis paradigm ,first an adversary generates an input instance , then this instance is randomly perturbed .the smoothed analysis paradigm was introduced by in 2001 to help explain why the simplex algorithm for linear programming works well in practice but not in ( worst - case ) theory .they considered instances formed by taking an arbitrary constraint matrix and perturbing it by adding independent gaussian noise with variance to each entry .they showed that , in this case , the shadow - vertex pivot rule succeeds in expected polynomial time .independently , studied the issue of hamiltonicity in a dense graph when random edges are added .in the context of graph optimization problems we can also mention , in the context of -sat , and in various other problems .our work joins this long series of papers studying perturbed instances in a variety of problems . specifically in our setting we study the following question : what if the balancers were not set completely adversarially but also not in a completely random fashion .besides the mathematical and analytical challenge that such a problem poses , in real network applications one may not always assume that the random source is unbiased , or in some cases one will not be able to quantitatively measure the amount of randomness involved in the network generation .still it is desirable to have an estimate of the typical behavior of the network .although we do not claim that our smoothed - analysis model captures all possible behaviors , it does give a rigorous and tight characterization of the tradeoff between the quality of load balancing and the randomness involved in setting the balancers directions , under rather natural probabilistic assumptions .as far as we know , no smoothed analysis framework was suggested to a networking related problem .formally , we suggest the following framework .we define both the smoothed - analysis aspect of the model , and the load - balancing one . for the load balancing part ,our model is similar to ( and , as we will shortly explain , a generalization of ) the periodic balancing circuits studied in .we think of the balancing network in terms of an -vertex graph . the processors in the network are the vertices of the graph , and balancers connecting processors are the ( directed ) edges of the graph .  before we proceed , since what follows is somewhat heavy on notation and indices , it will be helpful for the reader to bear in mind the following legend : we use superscripts ( in round brackets ) to denote a time stamp , and subscripts to denote an index .in subscripts , we use the vertices of the graph as indices ( thus assuming some ordering of the vertex set ) .for example , stands for the -entry in matrix , which corresponds to time / round .let be an arbitrary sequence of ( not necessarily perfect ) matchings .with each matching we associate a matrix with if and are matched in , if is matched in , if is not matched in , and otherwise .  in round ,every two vertices matched in perform a balancing operation .that is , the sum of the number of tokens in both vertices is split evenly between the two , with the remaining token ( if exists ) directed to the vertex pointed by the matching edge .  in * periodic balancing networks * ( see for example )an ordered set of ( usually perfect ) matchings is fixed .every round of balancing is a successive application of the matchings .our model is a ( slight ) generalization of the latter .let us now turn to the smoothed - analysis part . given a balancing network consisting of a set of directed matchings ,an * -perturbation * of the network is a flip of direction for every edge with probability independently of all other edges .setting gives the completely `` adversarial model '' , and is the uniform random case .for our results , it suffices to consider  ] , * moreover , for every wire , is a set of independent random variables .the proof of the lemma is given in section [ sec : proofofbalancersprop ] .let us remark at this point that the lemma holding under such weak conditions is rather surprising .first , it is valid regardless of the given orientation .secondly , the s of the balancers that affect the _ same _ output wire are independent . while this is obvious for balancers that are in the same layer , it seems somewhat surprising for balancers in subsequent layers that are connected .the remainder of the paper is organized as follows .we set out with the proof of theorem [ thm : lower ] in section [ sec : lower ] , preceding the proof of theorem [ thm : upper ] in section [ sec : upper ] .the reason is that the lower bound is concerned with the ( a special case of our general model ) .the techniques used in the proof of the lower bound serve as a good introduction to the more complicated proof of the upper bound . in sections [ sec : derivationofperiodiccase ] and [ sec : derivationofccc ] we show how to derive the special cases of the periodic balancing network and the from theorem [ thm : upper ] . finally we present experimental results that we obtain for the in section [ sec : conclusions ] .as we mentioned before , for the special case of the we adopt a `` topographical '' view of the network : calling the vertices _ wires _ , the time steps _ layers _ , the left - most side of the network the `` input '' and the right - most the `` output '' .the proof outline is the following . given an input vector ( uniformly distributed over the range ), we shall calculate the expected divergence from the average load .the expectation is taken over both the smoothing operation and the input .after establishing the `` right '' order of divergence ( in expectation ) we shall prove a concentration result .one of the main keys to estimating the expectation is lemma [ lem : balancersoddwithprobhalf ] saying that if the input is uniformly distributed as above , then for every balancer , }}=1/2 ] , ( c ) the simple fact that }}=\tfrac{1}{2}-\alpha ] for each . then for any number , } } \right| { \geqslant}\varepsilon\right ] } }     { \leqslant}2 \cdot \exp \left(-\frac{2 \varepsilon^2}{\sum_{i=1}^n ( b_i - a_i)^2 } \right).\ ] ]  for any random variable , let }} ] .the independence part follows from the fact that this is true for every conditioning on balancers from previous layers that affect ( by lemma [ lem : odd ] ) and the fact that balancers in the same layer that affect the same output wire are independent by construction , as those balancers depend on disjoint sets of balancers and disjoint sets of input wires .we prove the lemma by induction on , the layer of the balancer .the base case is immediate : the input to a balancer in layer is just the original input , which is by definition distributed uniformly over which is simply .assume the lemma is true for all balancers in layer and consider a balancer in layer .let be its two input wires . by the structure of the , the value on each wireis determined by a disjoint set of balancers and preceding wires , therefore we can treat , w.l.o.g . ,only .let be the part of that affects .( would be the set of balancers so that there is a path from them to .since the initial load on the input wires is chosen independently at every wire , and the sets of balancers affecting and are disjoint , indeed only affects , and the same applies for and ) .thus our goal is to calculate }}.\ ] ] let be the balancer in layer whose one outlet is , and let be its two inputs . recall that , and a possible + 1 addition in case this sum is odd and the balancer points in the direction of .furthermore it is easy to verify that for every ( assume is even , if odd write instead ) therefore for the event to occur , either  * , or * ( assuming w.l.o.g . that the balancer points towards , otherwise the sum equals ) .let us consider the first case . } }\\   & = { \ensuremath{\operatorname{\mathbf{pr}}\left[\bigvee_{i=0}^{n/2^{\ell-1}-1 } x_1 ' \equiv i \mod \left(n/2^{\ell-1}\right )    \wedge  x_2 ' \equiv 2k - i \mod \left(n/2^{\ell-1}\right ) \mid a_1(b)\right]}}.\end{aligned}\ ] ] now observe that the values of and are determined independently , as again , by the structure of the , they involve disjoint sets of balancers and input wires . similarly to we can define and which correspond to the parts of affecting and . by the structure of the , and on a disjoint set of input wires ( and balancers ) . as the input is chosen independently for every wire, does not affect and similarly does not affect .thus the latter reduces to }}\\   & \phantom{\sum_{i=0}^{n/2^{\ell-1}-1 } } \cdot { \ensuremath{\operatorname{\mathbf{pr}}\left[x_2'\equiv 2k - i \mod \left(n/2^{\ell-1}\right ) \mid a_2'(b)\right]}}.\end{aligned}\ ] ] by the induction hypothesis ( applied to the at layer ) , for every , each of }} ] is uniformly distributed over the range , and therefore in particular the entire expression does not depend on , or , put differently is the same for every choice of .the same argument holds for the case .this completes the proof .we shall derive our bound by measuring the difference between the number of tokens at any vertex and the average load ( as we did in the proof of the lower bound for the ) .specifically we shall bound , being the number of tokens at vertex at time ( we use for the vector of loads at time ) .there are two contributions to the divergence from ( which we analyze separately ) :  * the divergence of the idealized process from due to its finiteness . * the divergence of the actual process from the idealized process due to indivisibility .the idea to compare the actual process to an idealized one was suggested in and was combined with convergence results of markov chains .though we were inspired by the basic setup from and the probabilistic analysis from , our setting differs in a crucial point : when dealing with the case , we get a delicate mixture of the deterministic and the random model .for example , the random variables in our analysis are not symmetric anymore which leads to additional technicalities ( cf .lemma [ lem : independent ] ) .formally , let be the load vector of the idealized process at time , then by the triangle inequality ( is the all - one vector )   [ pro : triangleineq ] let be a balancing network with matchings .then ,  * , * over the -perturbation operation , +   theorem [ thm : upper ] then follows .the proof of the first part of the proposition consists of standard spectral arguments and is given in section [ sec : spectral ] for completeness .the proof the second part is more involved and is given in section [ sec : triangleineqtwo ] .letting be the initial load vector , it is easily seen that where is the matrix corresponding to matching ( as defined in section [ sec : themodel ] ) . for simplicitylet us abbreviate }:= { \ensuremath{\mathbf{p}}}^{(i)}\,{\ensuremath{\mathbf{p}}}^{(i+1)}\,{\ensuremath{\mathbf{p}}}^{(i+2)}\cdots { \ensuremath{\mathbf{p}}}^{(j)}.\ ] ] since } ] corresponding to .furthermore is just the total ( initial ) number of tokens , and therefore by definition we get .finally , let us project onto , that is , write ( as we said ) . for our goal to bound , it suffices to bound ( recall that refers to the -norm ) as for every vector , by the above , } \big\|\\   & = \bigg\| \sum_{i=2}^n c_i v_i \cdot { \ensuremath{\mathbf{p}}}^{[1,t_2 ] } \bigg\|\\   & = \bigg\| \sum_{i=2}^n c_i \lambda_i v_i \bigg\|.\end{aligned}\ ] ] recall that }) ] in absolute value . by the definition of the -norm , and using the fact that the form an orthogonal basis , the latter equals } ) \cdot \left(\sum_{i=2}^n c_i^2 \|v_i\|^2 \right)^{1/2}\\   = & \lambda({\ensuremath{\mathbf{p}}}^{[1,t_2 ] } ) \cdot \|\xi^{(0)}-\mu{\textbf{1}}\|\\   { \leqslant } &  \lambda({\ensuremath{\mathbf{p}}}^{[1,t_2]})\ , k { \hksqrt}{n}.\end{aligned}\ ] ]   the proof of this part resembles in nature the proof of theorem [ thm : lower ] . assuming an ordering of s vertices , for a balancer in round , , , we set if the initial direction ( before the perturbation ) is and otherwise ( in the lower bound we considered the all - up orientation thus we had no use of these variables ) . as in section[ sec : lower ] , for a balancer in round , the random variable is if the balancer is perturbed and otherwise . using these notationswe define a _ rounding vector _ , which accounts for the rounding errors in step .formally ,   now we can write the actual process as follows :    let be the set of balancers at time with no excess token , and the ones with. we can rewrite as follows :   unfolding equation , yields then } + \sum_{i=1}^t \rho^{(i ) } { \ensuremath{\mathbf{p}}}^{[i+1,t]},\ ] ] where } = \mathbf{i} ] is just ( as ) , and therefore } \\   & = { \textstyle\sum}_{i=1}^t       { \textstyle\sum}_{(u , v ) \in { \ensuremath{m^{(i)}_{{\mathsf{odd } } } } } }       \psi^{(i)}_{u , v } \cdot \phi^{(i)}_{u , v}\cdot(e_u - e_v)\cdot { \ensuremath{\mathbf{p}}}^{[i+1,t]}.\end{aligned}\ ] ] in turn , }_{u , w } - { \ensuremath{\mathbf{p}}}^{[i+1,t]}_{v , w}\right).\ ] ]  our next task is to bound equation  to receive the desired term from proposition [ pro : triangleineq ] .we do that similar in spirit to the way we went around in section [ sec : proofofpro : logn_lowerbound ] .we break this sum into its first summands ( whose expected sum we calculate and to which we apply a large - deviation - bound ) .the remaining terms are bounded deterministically .one major difficulty in the general setting is that lemma [ lem : balancersoddwithprobhalf ] ( which was crucial in the proof of theorem [ thm : lower ] ) does not hold in general as its proof makes substantial use of the special structure of the .equation [ eq : xminusxi ] with yields }_{u , w } -       { \ensuremath{\mathbf{p}}}^{[i+1,t_2]}_{v , w }       \big ) .    \end{aligned}\ ] ] with denoting the row - vector with at -th column and at -th column and zeros elsewhere , we can rewrite and split this equation as follows : } e_{w } \big ) \\       & & +       { \textstyle\sum}_{i = t_1 + 1}^{t_2 }       { \textstyle\sum}_{(u , v)\in { \ensuremath{m^{(i)}_{{\mathsf{odd } } } } } }       \psi_{u , v}^{(i ) } \ , \phi_{u , v}^{(i ) }       \ ,       \big ( e_{u , v } { \ensuremath{\mathbf{p}}}^{[i+1,t_2 ] } e_{w } \big ) .\end{aligned}\ ] ] clearly , } e_{w } \big ) \\      & = &  { \textstyle\sum}_{i = t_1 + 1}^{t_2 } \left (       { \textstyle\sum}_{(u , v)\in { \ensuremath{m^{(i)}_{{\mathsf{odd } } } } } }       \psi_{u , v}^{(i ) } \ , \phi_{u , v}^{(i ) }       \,e_{u , v }  \right ) \cdot \left (  { \ensuremath{\mathbf{p}}}^{[i+1,t_2 ] }  \ ,  e_{w }  \right ) .\end{aligned}\ ] ] observe that is a vector all of whose entries are bounded by in absolute value . since } ] , which is }  \ ,  e_{w}|_1 ] and , } } { \leqslant}(1/2-  \alpha). ] and } } \right| { \leqslant}t_1 \ , ( 1/2 -\ , \alpha ) ] hence by lemma [ lem : independent ] we obtain with }}| + { \hksqrt}{\gamma \ , \log n} ] and }}| ] , + and for any number , } } { \geqslant}{\ensuremath{\operatorname{\mathbf{pr}}\left [ x < - \delta\right]}} ] , + and for any number , } } { \geqslant}{\ensuremath{\operatorname{\mathbf{pr}}\left [ x > \delta\right]}} ] . since for any , stochastically dominates , we obtain } } { \leqslant}{\ensuremath{\operatorname{\mathbf{pr}}\left[y < 0\right ] } } { \leqslant}1/2 ,    ] 2 .  if all , then }}\right ] } } { \leqslant}7/8 . ] , as otherwise is symmetrically distributed around .finally , we also assume the weights to be integral , that is , .rational weights can be easily reduced to integral weights by multiplying all weights with their least common multiple of the denominators and applying the bound for the integral case .  for the sake of contradiction ,suppose that }}\right ] } } > 7/8 ] and hence } } { \leqslant}{\ensuremath{\operatorname{\mathbf{pr}}\left [ x { \leqslant}2 { \ensuremath{\operatorname{\mathbf{e}}\left[x\right]}}\right ] } } < 1/8 ] be such that } } { \geqslant}1/8, ]  as we assumed the weights to be integral , we can use the following two well - known counting tricks : } }       & = \sum_{x{\geqslant}k } \sum_{y=1}^x { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right]}}\\       & = \sum_{y{\geqslant}1 } \sum_{x{\geqslant}\max(k , y ) } { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right]}}\\       & = \sum_{x{\geqslant}1 } { \ensuremath{\operatorname{\mathbf{pr}}\left[x{\geqslant}\max(k , x)\right ] } } ,     \end{aligned}\ ] ] and similarly , } } = \sum_{x{\geqslant}1 } { \ensuremath{\operatorname{\mathbf{pr}}\left[x{\leqslant}-\max(k , x)\right]}} ] , we get } } & = \sum_{x } x { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right]}}\\       & = \sum_{x{\leqslant}-1 } x { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right ] } } +        \sum_{0{\leqslant}x < k } x { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right ] } } +        \sum_{x{\geqslant}k } x { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right ] } } \\       & { \geqslant}k\,({\ensuremath{\operatorname{\mathbf{pr}}\left[x{\geqslant}k\right]}}-1/8 ) + \sum_{0{\leqslant}x < k } x { \ensuremath{\operatorname{\mathbf{pr}}\left[x = x\right]}}.   \end{aligned}\ ] ] using above assumptions on and }}\right ] } } > 7/8 ] be the set of all matching edges in the given time span .let }_{{\mathsf{odd}}}}} ] .we now begin by bounding in terms of . by the definition of conditional probability and the law of total probabilities, it follows for arbitrary , } }   =   \frac     { { \ensuremath{\operatorname{\mathbf{pr}}\left[w<\delta_2 \wedge w_{{\mathsf{odd}}}{\geqslant}\delta_1\right ] } } }     { { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\geqslant}\delta_1\right]}}}\notag\\   & { \leqslant}\frac     { { \ensuremath{\operatorname{\mathbf{pr}}\left[w - w_{{\mathsf{odd}}}<\delta_2-\delta_1 \wedge w_{{\mathsf{odd}}}{\geqslant}\delta_1\right ] } } }     { { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\geqslant}\delta_1\right]}}}\notag\\   & =   \sum_{{\ensuremath{\mathcal{m } } } }     \frac {       { \ensuremath{\operatorname{\mathbf{pr}}\left[{\ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } }       { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{even}}}<\delta_2-\delta_1 \wedge       w_{{\mathsf{odd}}}{\geqslant}\delta_1 \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } }     } { { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\geqslant}\delta_1\right ] } } } \notag.\end{aligned}\ ] ]  note that for fixed , the ranges of and are determined and therefore the probability spaces are disjoint .this implies that and are independent conditioned on }_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}} ] and in ( * ) we obtain } } \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \notag\\   & \qquad= 1 - { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{even } } } { \geqslant}2 \ , { \ensuremath{\operatorname{\mathbf{e}}\left[w^{-}\right ] } } \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \notag \\   & \qquad{\leqslant}1 - { \ensuremath{\operatorname{\mathbf{pr}}\left [ w^+_{{\mathsf{even } } } { \geqslant}0 \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \notag\\   & \qquad\phantom{{\leqslant}}\cdot { \ensuremath{\operatorname{\mathbf{pr}}\left [ w^-_{{\mathsf{even } } } { \geqslant}2 \ , { \ensuremath{\operatorname{\mathbf{e}}\left[w^{-}\right ] } } \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right]}}. \notag\end{aligned}\ ] ]  observing that } } = { \ensuremath{\operatorname{\mathbf{e}}\left[\psi_{u , v } \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right]}} ] and hence by lemmas [ lem : simplelemma ] and [ lem : negrattail ] , } } \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \notag \\   & \qquad{\leqslant}1 - { \ensuremath{\operatorname{\mathbf{pr}}\left [ w^+_{{\mathsf{even } } } { \geqslant}0 \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \cdot \notag\\   & \qquad\qquad{\ensuremath{\operatorname{\mathbf{pr}}\left [ w^-_{{\mathsf{even } } } { \geqslant}2 \ , { \ensuremath{\operatorname{\mathbf{e}}\left[w^{-}_{{\mathsf{even } } } \ , \mid \ , { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } } \notag \\   & \qquad{\leqslant}1 - \tfrac{1}{2 } \cdot \tfrac{1}{8 }   = \tfrac{15}{16}.   \label{eq : applytail}\end{aligned}\ ] ] plugging this into equation  yields } }  \mid w_{{\mathsf{odd}}}{\geqslant}\delta\right ] } } \\   & \qquad{\leqslant}\frac{15}{16 } \ , \sum_{\ensuremath{\mathcal{m}}}\frac {       { \ensuremath{\operatorname{\mathbf{pr}}\left[{\ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } }         { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\geqslant}\delta_1 \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } }     } { { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\geqslant}\delta_1\right ] } } } = \frac{15}{16},\end{aligned}\ ] ] and hence }}\right]}}\notag\\   & \qquad{\geqslant}{\ensuremath{\operatorname{\mathbf{pr}}\left [ w { \geqslant}\delta + 2 { \ensuremath{\operatorname{\mathbf{e}}\left[w^-\right ] } } \wedge w_{{\mathsf{odd } } } { \geqslant}\delta\right ] } } \notag\\   & \qquad= { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd } } } { \geqslant}\delta\right ] } } \cdot { \ensuremath{\operatorname{\mathbf{pr}}\left[w { \geqslant}\delta + 2 { \ensuremath{\operatorname{\mathbf{e}}\left[w^-\right ] } }    \ , \mid \ ,w_{{\mathsf{odd } } } { \geqslant}\delta\right ] } } \notag\\   & \qquad{\geqslant}\tfrac{1}{16 } \cdot { \ensuremath{\operatorname{\mathbf{pr}}\left [ w_{{\mathsf{odd } } } { \geqslant}\delta\right]}}. \label{eq : firststep}\end{aligned}\ ] ]  it remains to lower bound the deviation of in terms of in a similar fashion . as before , we derive } } \notag\\[-.3 cm ]   & \qquad{\leqslant}\sum_{\ensuremath{\mathcal{m}}}{\ensuremath{\operatorname{\mathbf{pr}}\left[{\ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } }     \overbrace{{\ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{even}}}>\delta_2-\delta_1 \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right]}}}^ { ( * ) }     \label{eq : thomasfavorite2}\\[-.1 cm ]     & \qquad \qquad \qquad \cdot{\ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\leqslant}\delta_1 \mid { \ensuremath{m^{[t_1\!,t_2]}_{{\mathsf{odd}}}}}={\ensuremath{\mathcal{m}}}\right ] } }      \big/ { \ensuremath{\operatorname{\mathbf{pr}}\left[w_{{\mathsf{odd}}}{\leqslant}\delta_1\right]}}.\notag\end{aligned}\ ] ] we now choose }}, ] to gives the assertion of the lemma .consider the network after ( periodic ) repetitions of the balancing network ( each repetition consists of rounds , so the network consists of a total of matchings ) . is the so - called round matrix corresponding to the application of consecutive matchings ( recall that we use the following abbreviations }={\textstyle\prod}_{i = t_1}^{t_2 } { \ensuremath{\mathbf{p}}}^{(i)} ] and so on .recall the notation that denotes the row - vector with at -th column and at -th column .in theorem [ thm : upper ] we plug and .hence , }e_w\right)^2}.   \end{aligned}\ ] ] let be the eigenvectors of } ] is a stochastic matrix , .write . using this ,we rewrite as follows ( for short we drop the `` '' part as our final result will not depend on ) , } \sum_{j=1}^n c_j^{(i ) } v_j^{(i ) } \right)^2 } \\     & = { \hksqrt}{\log n \ , { \textstyle\sum}_{i=1}^{t_1 }       { \textstyle\sum}_{(u , v)\in { \ensuremath{m^{(i ) } } } }       \left(e_{u , v } \sum_{j=1}^n c_j^{(i ) } \lambda_j^{(i ) } v_j^{(i ) }  \right)^2}.  \end{aligned}\ ] ] observe that is orthogonal to .therefore for every which gives define a vector .the latter is then since is a matching , for each each vertex is counted only at most once , thus   by standard calculation ( cf .section [ sec : spectral ] ) , we have })^2 \cdot \| e_w \|^2 = \lambda({\ensuremath{\mathbf{p}}}^{[i+1,t_2]})^2 .   \end{aligned}\ ] ] plugging this into ( [ eq : letzter ] ) and using the facts that for any two stochastic matrices and , and for any integer , , })^2 } \\   & { \leqslant}2 { \hksqrt } { \log n \ , { \textstyle\sum}_{i=1}^{t_1 } \lambda ( { \ensuremath{\mathbf{q}}}^{\lfloor ( t_2-(i+1))/d \rfloor } ) ^2}.   \end{aligned}\ ] ] regrouping the matrices according to the periods the latter reduces to with , we can upper bound by plugging in our choices for and we end up with   with the same arguments , we get } ) \ , { \hksqrt}{n } k { \leqslant}\ , { \hksqrt}{n } k/ ( kn)^2 .\end{aligned}\ ] ] plugging all this into theorem [ thm : upper ] shows that at step the discrepancy is at most theorem [ thm : upper ] we choose and . for matrix multiplicationwe use the following abbreviated form : }:= { \ensuremath{\mathbf{p}}}^{(i)}{\ensuremath{\mathbf{p}}}^{(i+1)}\cdots { \ensuremath{\mathbf{p}}}^{(j)} ] if wires and are at distance and differ only in the last bits ( in their binary representation ) , * otherwise , }_{u , v}= 0 ] is equal to the all- matrix and thus } \right)=0 $ ] ( that is , ) .moreover , for any fixed wire , } \right ) e_w     \right)^2 }     \\     & =      { \hksqrt } { \log n\,{\textstyle\sum}_{i=1}^{\log n - \log \log n }  { \textstyle\sum}_{(u , v ) \in { \ensuremath{m^{(i ) } } } }     \left ( { \ensuremath{\mathbf{p}}}^{[i+1,t_2]}_{u , w } - { \ensuremath{\mathbf{p}}}^{[i+1,t_2]}_{v , w }     \right)^2 }      \\     & { \leqslant}{\hksqrt } { \log n \ , { \textstyle\sum}_{i=1}^{\log n - \log \log n } \frac{n}{2^{i } } \left ( \frac{2^{i+1}}{n } \right)^2 } \\     & = 4 \ , { \hksqrt } { \log n \ , { \textstyle\sum}_{i=1}^{\log n - \log \log n } \frac{2^i}{n } } \\     & { \leqslant}4 \ , { \hksqrt } { \log n\cdot \frac{2}{\log n } } \\     & = { \mathcal{o}}(1).\end{aligned}\ ] ]  -values of with random uniformly distributed input from . corresponds to the adversarial model while is the completely random model .the dotted line describes the experimental results , the broken lines are our theoretical lower and upper bounds.,scaledwidth=66.0% ]we examined experimentally how well a  balances a random input .we implemented a consisting of roughly one billion wires and thirty billion balancers .the input was chosen independently uniformly at random from . for initial directions of the balancers all up and different values between and we measured the resulted discrepancy .figure [ fig : experiments ] presents the average over 100 runs , together with theoretical lower and upper bounds . as the input is random ,the so - far presented bounds can be tightened .following the same lines of proof , one can easily show the following slightly better bounds on the _ expected _ discrepancy in the random - input case :     d. arthur and s. vassilvitskii .worst - case and smoothed analysis of the icp algorithm , with an application to the k - means method . in _47th ieee symp . on found . of comp .science ( focs06 ) _ , pages 153164 , 2006 .a. coja - oghlan , u. feige , a. m. frieze , m. krivelevich , and d. vilenchik . on smoothed -cnf formulas and the walksat algorithm . in _20th acm - siam symp . on discrete algorithms ( soda09 )_ , pages 451460 , 2009 .t. friedrich , t. sauerwald , and d. vilenchik . smoothed analysis of balancing networks . in _36th international colloquium on automata , languages , and programming ( icalp09 ) _ , volume 5556 of _ lecture notes in computer science _ , pages 472483 .springer , 2009 .y. rabani , a. sinclair , and r. wanka . local divergence of markov chains and the analysis of iterative load balancing schemes . in _39th annual ieee symposium on foundations of computer science ( focs98 ) _ , pages 694705 , 1998 .

Abstract: in a balancing network each processor has an initial collection of unit - size jobs ( tokens ) and in each round , pairs of processors connected by balancers split their load as evenly as possible . an excess token ( if any ) is placed according to some predefined rule . as it turns out , this rule crucially affects the performance of the network . in this work we propose a model that studies this effect . we suggest a model bridging the uniformly - random assignment rule , and the arbitrary one ( in the spirit of smoothed - analysis ) . we start with an arbitrary assignment of balancer directions and then flip each assignment with probability independently . for a large class of balancing networks our result implies that after rounds the discrepancy is with high probability . this matches and generalizes known upper bounds for and . we also show that a natural network matches the upper bound for any .


in 1998 hubble plots of distant type ia supernovae revealed that the universe began accelerating somewhat over six billion years ago .supernova data since then have continued to support this conclusion and it has been confirmed by other data sets .this triumph of observational cosmology continues to challenge fundamental theorists .  the data are consistent with general relativity operating on a critical energy density whose current composition is about 70% cosmological constant , with about 30% nonrelativistic and small amounts of relativistic matter . however , fundamental theorists have a hard time understanding why the cosmological constant should be much smaller than any other scale in physics and why it should have the right magnitude to so recently come into dominance .scalar potential models can be devised to reproduce the observed expansion history but they are difficult to motivate .quantum effects from a very light scalar have also been suggested .modifications of gravity have been considered . however , generalizing the hilbert lagrangian from to  represents the only local , metric - based , generally coordinate invariant and stable modification of gravity . because the only model within this class which exactly reproduces the expansion history is general relativity with  , more general models can deviate from observation even without considering perturbations .there is greater freedom to modify gravity if locality is abandoned , but one should keep in mind newton s famous dictum of 1692/3 :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ that one body may act upon another at a distance thro a vacuum , without the mediation of any thing else , by and through which their action and force may be conveyed from one to another , is to me so great an absurdity that i believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it . __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  the great man s misgivings have certainly stood the test of time : more than three centuries of observation and experimentation have failed to reveal a single exception to the paradigm of local , second order field equations . proposed theoretical exceptions , such as string field theory , tend to have extra degrees of freedom which carry negative energy .my view is that nonlocal modifications of gravity can not be fundamental .if these models describe nature , i believe their nonlocal structures must derive from the gravitational vacuum polarization of infrared gravitons which were produced during the epoch of primordial inflation .this is a natural expectation because accelerated expansion creates a vast ensemble of infrared gravitons it causes the tensor power spectrum  and because the large vacuum energy of primordial inflation provides a dimension three self - interaction .of course loops of massless particles can produce long range effects , as witness confinement in quantum chromodynamics ( qcd ) .gravitons with zero vacuum energy have no impact on cosmology merely because the self - interaction has dimension five . however , a dimension three coupling between massless particles ought to engender stronger infrared effects than the dimension four couplings of cqd .quantum gravitational self - interactions during inflation are suppressed by the small ( ) loop - counting parameter of quantum gravity , but they also grow with the number of inflationary e - foldings . the growth occurs because more and more infrared gravitons come into causal contact the longer inflation persists .one consequence of it is that the interactions become nonperturbatively strong during a sufficiently prolonged period of inflation .that is frustrating because it means some sort of resummation technique must be devised in order to gain quantitative control over late time cosmology .this task is not hopeless it can be done for scalar models  but it has not been accomplished yet for quantum gravity . in the absence of such a nonperturbative resummation techniquethe approach i shall describe here is the purely phenomenological one of proposing a plausible set of nonlocal field equations and examining their properties .however , it is well to note that models which might eventually be derived from inflationary vacuum polarization must possess two features which would be forbidden in fundamental theory :  * there is an initial value surface corresponding to inflation s end ; and * because the putative effect derives from cosmological - scale gravitons it is completely natural that there should no change in gravity on small scales .the model i will describe in this article was developed with stanley deser .we change the gravitational lagrangian from that of general relativity by the addition of a nonlocal term of the form , here  ] must be _ conserved _ , that is their covariant divergence must vanish , . that follows automatically from varying a diffeomorphism invariant action , whether the lagrangian is local or nonlocal .the nonlocal corrections we seek must also be _causal_. that is , the variation of (x) ] could be viewed as a sort of time - varying newton constant , with the remaining terms there to enforce conservation .the arbitrary function in expression ( [ dl2 ] ) is known as the _ nonlocal distortion function_. many models of dark energy and modified gravity have such free functions .the _ reconstruction problem _ consists of adjusting these functions to reproduce a given expansion history , which is usually .the geometry is homogeneous , isotropic and spatially flat , for the reconstruction problem we assume and all its derivatives are known functions of time .it is straightforward to solve the reconstruction problem for scalar quintessence models and a brief presentation of the solution will help fix ideas . for the geometry ( [ flrw ] )the scalar depends just on time and only two of einstein s equations are nontrivial , \ ; , \label{e1 } \\-2 \dot{h } - 3 h^2 & = & 8\pi g \bigl [ \frac12 \dot{\varphi}^2 -  v(\varphi)\bigr ] \ ; . \label{e2}\end{aligned}\ ] ] as usual , is the newton constant , and a dot denotes the time derivative . by adding ( [ e1 ] ) and ( [ e2 ] ) one obtains the relation , hence one can reconstruct the scalar s evolution provided the hubble parameter is monotonically decreasing , this relation can be inverted ( numerically if need be ) to give the time as a function of the scalar , .one then determines the potential by subtracting ( [ e2 ] ) from ( [ e1 ] ) and evaluating the resulting function of time at , \ ; .\ ] ]  i will review a similar construction for the nonlocal distortion function that was devised with cedric deffayet . it and like techniques allow one to reproduce the expansion history or any other without a cosmological constant or dark energy .this is an important distinction between nonlocal cosmology and models .  for the geometry ( [ flrw ] ) the nontrivial field equations are , here and are the energy density and pressure , which are assumed to be known functions of time .for example , to reproduce the expansion history without we would use , \ ; , \\p(t ) & = & \frac{3 h_0^ 2}{8 \pi g } \bigl [ 0 + \frac13 \omega_{\rm rad } \bigl (  \frac{a_0}{a(t ) } \bigr)^4 \bigl ] \ ; , \end{aligned}\ ] ] where and are the current values of the scale factor and the hubble parameter , and and are the fractions of the current energy density in matter and radiation .the specialization of the nonlocal modification terms to the geometry ( [ flrw ] ) gives ,  \biggl\ { f \bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) + \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } }  \biggl [ r f'\bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr)\biggr ] \bigg\ } } \nonumber \\ & & \hspace{4 cm } + \frac12 \partial_{t } \bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) \times  \partial_{t } \biggl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } \biggl [ r f'\bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r  \bigr ) \biggr ] \biggr ) , \qquad \label{d200 } \\\lefteqn{\delta g_{ij } = -\bigl[2 \dot{h } + 3 h^2 + 2 h \partial_{t } +  \partial_{t}^2\bigr ] \biggl\ { f\bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) +  \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } \biggl [ r f'\bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr)\biggr ] \bigg\ }  g_{ij } } \nonumber \\ & & \hspace{4 cm } + \frac12 \partial_{t } \bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) \times  \partial_{t } \biggl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } \biggl [ r f'\bigl(\frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r  \bigr ) \biggr ] \biggr ) g_{ij } \ ; .\qquad \label{d2ij}\end{aligned}\ ] ] for this geometry the ricci scalar is and the action of on some function of time can be expressed as a double integral from the initial time , (t ) = -\int_{t_i}^t \!\ ! dt ' \ , \frac1{a^{3}(t ' ) } \int_{t_i}^{t ' } \!\ !dt '' \ , a^3(t '' ) \ , w(t '' ) \ ; . \label{invbox}\] ]    the first step of our construction is taking the difference between relations ( [ eqrho ] ) and ( [ eqp ] ) .this leads to a simple equation for the function , \qquad , \qquad x(t ) \equiv \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width0.6pt}\hrule height 0.6pt}\kern1pt}}}\bigl [ r\bigr](t ) \ ; . \label{ff2}\ ] ] this differential equation is , recognizing that is a homogeneous solution permits us to reduce the order so that the solution can be expressed in terms of an integral . changing the dependent variable to us to express ( [ ode ] ) as , = a^5 \bigl [ 8\pi g ( \rho - p ) - ( 2\dot{h }  + 6 h^2 ) \bigr ] \equiv a^3(t ) \times s(t ) \ ; . \label{ode2}\ ] ] the solution to ( [ ode2 ] ) for general can be written as a double integral , requiring no deviation from general relativity at the initial time corresponds to the initial values , which allows us to express the function using the notation ( [ invbox ] ) , (t ) \ ; . \label{fsol}\ ] ]  the next step is to act on relation ( [ ff2 ] ) and regard the result as a differential equation for the function , we can find an integrating factor for ( [ ode3 ] ) in terms of the function , multiplying ( [ ode3 ] ) by gives a form which can be immediately integrated , = e^{z(t ) }  \frac{d}{dt } \bigl [ a^3(t ) \ , f(t ) \bigr ] \ ; .\ ] ]  the penultimate step is to solve for time as a function of , the desired nonlocal distortion function is evaluated at , all of these steps are straightforward to implement numerically for any scale factor . fig [ graph ] shows the nonlocal distortion function that deffayet and i constructed for the values pertinent to the five - year wmap data .( dodelson and park have carried out the procedure with current data . )there is a simple analytic form for which is indistinguishable from the numerical solution , \quad , \quad y \equiv x + 16.5 \ ; . \label{analytic}\ ] ] in fact the current data on the expansion history do not require keeping the higher powers of in the argument of the function .modified gravity models based on changing to suffer from a major problem in that typically has the same sign for cosmology , where we want big effects to explain the acceleration data , and for the solar system , where significant deviations from general relativity are not permissible .this has prompted the development of elaborate `` chameleon mechanisms '' in which the extra scalar degree of freedom present in models is light in cosmological settings and heavy inside the solar system .nonlocal cosmology differs from models in two crucial ways :  * there are no extra degrees of freedom to mediate new forces ; and * the factor of acting on allows us to define the nonlocal distortion function so that there are no changes from general relativity _ at all _ in a gravitationally bound system , without affecting cosmology .the first point will be demonstrated in section [ dof ] ; it is the second point which concerns us here .the key fact is that the scalar dalembertian has different signs when acting on functions of time than on functions of space .this is obvious from the flat space limit . in the background cosmology , and small perturbations about it , the time dependence of the ricci scalar is stronger than its dependence upon space .this means that is typically negative for cosmology .that is clear from the minus sign in expression ( [ xdef ] ) .indeed , the reconstruction procedure of section [ recon ] determines the nonlocal distortion function only for . this can be seen from the graph in fig .[ graph ] : reconstruction does not fix the form of for .although gravitationally bound systems are not always static , it is generally true that their dependence upon space is stronger than their dependence upon time .that means is positive inside a gravitationally bound system .further , reproducing the expansion history requires  . if we wish to completely eliminate any corrections inside gravitationally bound systems it will suffice to define for all .hence there is a very simple way to make screening 100% effective inside the solar system , the galaxy , or any other gravitationally bound system , all without affecting the model s behavior for cosmology .some model builders consider that requiring to vanish for represents a horrendous amount of fine - tuning. this objection would be tenable if one regarded nonlocal cosmology as some sort of fundamental theory , but that is precisely what it can not be . the initial time we saw in expressions ( [ invbox ] ) , ( [ phi ] ) , ( [ zdef ] ) and ( [ xdef ] ) has no place in a fundamental theory .if the model is to avoid newton s famous philosophical injunction  and more modern pitfalls  its nonlocal modifications must represent the most cosmologically significant quantum gravitational corrections to the effective field equations from the vast ensemble of infrared gravitons produced during primordial inflation .those gravitons were of _ horizon scale _ when they were ripped from the vacuum .( that is why one can study the process using quantum general relativity . )it makes perfect sense for them to have the biggest effect on very large scales , and little or no effect on small scales .this is not exquisite fine - tuning but rather an inevitable and completely natural consequence of the putative physical origin of nonlocal cosmology .the past two sections have shown how the nonlocal distortion function can be defined to reproduce the expansion history , without changing the predictions of general relativity for gravitationally bound systems .we seek now to understand what sort of initial value formalism the model possesses .in particular , what are its degrees of freedom and constraints ? and do any of its degrees of freedom ever become ghosts ? these issues were settled by a recent paper written with stanley deser .   shortly after the nonlocal cosmology model ( [ dl2 ] ) was proposed nojiri and odinstov devised a local version based on two scalar fields .their idea is for variation with respect to a scalar to enforce the equation , so that becomes plus an arbitrary homogeneous solution .the nonlocal distortion terms in ( [ dl2 ] ) are subject to the modification , this localized version of the model contains two scalar degrees of freedom which are not present in the original , nonlocal model ( [ dl2 ] ) .the purpose of this subsection is to demonstrate that one of these extra degrees of freedom is a ghost , as pointed out to me by gilles esposito - farese .  to exhibit the ghost ,first partially integrate the scalar kinetic term of ( [ locver ] ) .next express this as a difference of two squares , with our spacelike signature one can see that the combination has positive kinetic energy whereas the combination is a ghost .  the analysis i have just given is valid so long as the metric is any nondynamical background .but our metric is of course dynamical that is the point of modified gravity ! and one might worry about differentiated metric fields from the term mixing with the other fields to stabilize the ghost .first , note that such a stabilization would not be phenomenologically acceptable , if it did occur , because would still develop large inhomogeneities because is a ghost in _ any _ background metric .however , the question was analysed in detail by nojiri , odintsov , sasaki and zhang , who concluded that a ghost would form unless , the authors asserted that classical evolution from special initial conditions could result in these inequalities being preserved for long periods .that is true enough , but it ignores the virulence of kinetic instabilities .kinetic instabilities are driven by the infinite volume of phase space in the ultraviolet , and they will lead to instantaneous decay if any initial value data permit them . the fact that condition ( [ cond ] ) is _ linear _ in the independent field means that the local model can not be stable .it might be worried that the existence of this unstable scalar extension of the original model ensures that nonlocal cosmology is itself unstable .of course the original , nonlocal model is just the local one subject to the initial value constraints .why , it might be wondered , can we not excite the ghost degree of freedom ?however , general relativity offers an example in which constraining a particular variable converts an unstable model into a stable one .the variable we have in mind is of course the newtonian potential , which would be a ghost , and fatal to the theory , were it not a constrained field .so it is at least conceivable that the original , nonlocal model ( [ dl2 ] ) is stable . in the remainder of this sectioni will demonstrate that it is at indeed free of ghosts .synchronous gauge is the frame of freely falling observers who are released from a spacelike surface with zero relative velocities , note that denotes the full 3-metric , including the cosmological background of .synchronous gauge has well - known problems with caustics which should not be important for our purposes .we believe that the basic analysis and conclusions of this section would apply for any lapse and shift .  in synchronous gaugethe covariant scalar dalembertian takes the form , here and henceforth , denotes the inverse of the spatial metric , stands for the determinant of , and an overdot represents differentiation with respect to time .the various curvatures we require are , note that we follow the usual convention in which a superscript before a quantity denotes its specialization to the purely spatial geometry .let us first see that the nonlocal field equations require the same initial value data as general relativity , namely , the values of the 3-metric and its first time derivative at : and . the retarded green s function associated with is defined by the differential equation , (t,\vec{x};t',\vec{x } ' ) = \delta(t \!-\ !t ' ) \delta^3(\vec{x } \!-\ ! \vec{x } ' ) \ ; , \label{geqn}\ ] ] subject to retarded boundary conditions , (0,\vec{x};t',\vec{x } ' ) = 0 \qquad \forall \ , t ' > t \ ; . \label{gbc}\ ] ] even though we can not solve equations ( [ geqn]-[gbc ] ) for an arbitrary 3-metric , their form clearly defines the green s function  ] multiplied by the ricci scalar , we need only consider the second time derivatives of the ; the lower time derivatives and all spatial derivatives are shielded by the inverse differential operator . from expression ( [ ricci ] ) we see that these second time derivatives can be written in form , now use relation ( [ dalem ] ) to express second time derivatives in terms of , we can obviously combine relation ( [ dtform ] ) with ( [ rform ] ) to conclude , here is the spatial affine connection and a comma denotes partial differentiation.  with relations ( [ geqn]-[gbc ] ) , equation ( [ rbox ] ) shows that involves only the usual initial value data of general relativity : and .we can show that these initial value data are apportioned the same way ( as general relativity ) between constrained fields and gravitational radiation by examining the nonlocal corrections and to the constraint equations .note first from ( [ gbc ] ) that and its first time derivative both vanish at .further , the nonlocal distortion function vanishes at .so we need only examine the two terms of in which two covariant derivatives act upon  ] .the latter terms are simple to analyze , the local second derivative terms are therefore , \biggr ] } \nonumber \\ & & \hspace{-.5 cm } - \frac12 h_{ij } h^{k\ell } \ddot{h}_{k\ell } \times \biggl [ 1 + f\bigl ( \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) + \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } }  \bigl[r f'\bigl ( \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) \bigr ] - 4 f'\bigl (  \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r\bigr ) \biggr ] + o(\partial_t ) \ ; .\qquad \label{nonkey}\end{aligned}\ ] ] only the first line of expression ( [ nonkey ] ) represents the unconstrained , dynamical part of . by comparing with the approximate analytic form ( [ analytic ] ) of the nonlocal distortion function we see that the coefficient of the dynamical term is reduced at late times , but never by enough to make it change sign .we therefore conclude that no dynamical graviton mode ever becomes a ghost .midway through the reconstruction work with cedric deffayet it became obvious to me that models of the type ( [ dl2 ] ) were likely to have problems with structure formation because of how they contrive to support late time acceleration without dark energy . to see the problem ,specialize the friedmann equation of general relativity to an energy density comprised of only nonrelativistic matter , dark energy was posited because the data says the left hand side of ( [ friedmann ] ) is approaching a constant whereas the right hand side is falling off . to understand how nonlocal cosmology resolves this conundrum it suffices to specialize the nonlocal corrections ( [ mod2 ] ) to slowly varying and nearly homogeneous geometries . under these two assumptionsone can drop all the derivatives , \biggl\ { f\bigl ( \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r \bigr ) + \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } \biggl [ r f'\bigl ( \frac1{{{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt } } } r \bigr ) \biggr ] \biggr\ } \ ; . \label{simpdg}\ ] ] of course expression ( [ simpdg ] ) is proportional to the einstein tensor , which means that , for slowly varying and nearly homogeneous geometries , we can regard the modified gravity equations as those of general relativity with a time - varying effective newton constant , } \ ; . \label{geff}\ ] ] the right hand side of the modified friedmann equation ( ) is brought into balance with the nearly constant left hand side because the growth in compensates for the redshift of the energy density .  _ strengthening the force of gravity is dangerous ! _the fact that it happens for models of the form ( [ dl2 ] ) convinced me , back in the fall of 2008 , that this class of models can not be correct .however , the problem is not as bad as one might imagine .for example , there is nothing wrong inside the solar system where we have very strong constraints on time variation in newton s constant . for any gravitationally bound systemthe geometry is not homogeneous so the approximation ( [ simpdg ] ) is invalid . in any casethe effective newton constant ( [ geff ] ) degenerates to inside any gravitationally bound system because and vanishes for positive , as we saw in section [ screen ] . to encounter a real problem one must study the regime in which perturbations around the cosmological background are smallthat is what dodelson and park did .i wo nt attempt to reproduce their work , but i will explain some of the preliminaries and summarize their conclusions .dodelson and park consider scalar perturbations to ( [ flrw ] ) , dt^2 + a^2(t ) \bigl [ 1 + 2 \phi(t,\vec{x } ) \bigr ] \ ; . \label{potentials}\ ] ] they treat the potentials and to be small spatial plane waves , they then work out the first order perturbation to the nonlocally modified equation for nonrelativistic matter , it helps to see the purely temporal and spatial parts of , -2 \delta^{ij } \bigl [ \ddot{\widetilde{\phi } } + 3 h \dot{\widetilde{\phi } } + h \dot{\widetilde{\psi } } - ( 2\dot{h } + 3 h^2 ) \widetilde{\psi } \bigr ] .\qquad \label{gij}\end{aligned}\ ] ] because expression ( [ gij ] ) contains two independent tensor factors , we can infer three equations for the three variables , and .the nonlocal corrections do not change this overall structure , although they do alter important things such as the general relativistic relation between the two potentials .i do wish to explain how to perturb functions of  , , \label{key1}\ ] ] where , .\label{key2}\ ] ] the retarded green s function , \ ; , \ ] ] is formed from the mode functions of the massless , minimally coupled scalar , for the subhorizon modes of interest , dodelson and park were able to ignore the various time derivatives in expression ( [ key2 ] ) and they employed the wkb approximation for the retarded green s function , \ ; .\ ] ]  dodelson and park found only small changes in the potential , however , they discovered that the newtonian potential grows rapidly from about redshift until it is nearly double that of general relativity at the current time .this is what one expects from a model which strengthens the gravitational force , but it is a phenomenological disaster : weak lensing favors general relativity over nonlocal cosmology by almost , and redshift space distortions favor general relativity by almost  .particle theorists are notorious for concocting rosy interpretations of the most devastating phenomenology . at the risk of further damaging our already abysmal reputation ,let me say that the recent paper by dodelson and park does not depress me , nor does it particularly surprise me .their excellent work rules out the simplest class of models ( [ dl2 ] ) , but i ceased taking those models seriously after realizing that they achieve cosmic acceleration by making the effective newton constant grow . what one wants is a nonlocal model in which the cosmological constant becomes time dependent , and i eventually succeeded in devising a class of them in collaboration with nick tsamis seems clear that an elaboration of these models can reproduce the expansion history without disastrously enhancing early structure formation the way the simple models ( [ dl2 ] ) do .in fact the 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Abstract: i review a class of nonlocally modified gravity models which were proposed to explain the current phase of cosmic acceleration without dark energy . among the topics considered are deriving causal and conserved field equations , adjusting the model to make it support a given expansion history , why these models do not require an elaborate screening mechanism to evade solar system tests , degrees of freedom and kinetic stability , and the negative verdict of structure formation . although these simple models are not consistent with data on the growth of cosmic structures many of their features are likely to carry over to more complicated models which are in better agreement with the data .


transition path sampling ( tps ) is a successful approach to characterizing rare event processes for which _ a priori _ knowledge of the location of , or even existence of , a single well - defined transition state is not available .it addresses the rare event problem by focusing computational resources on the small subset of trajectory space that exhibits dynamically interesting events .the development of the approach into a computationally effective tool is due largely to the efforts of chandler and coworkers, and it is now widely used to study the mechanisms and rates of reactive processes .the tps method prescribes metropolis monte carlo sampling of the transition path ensemble , which includes only those dynamical trajectories of a given length in time that begin in a prescribed `` reactant '' region and terminate in prescribed `` product '' region .a central feature of the tps method is the preservation of the boltzmann distribution through detailed balance .  as in all metropolis algorithms ,tps involves a proposal step , in which a new dynamical trajectory is constructed , and then an acceptance / rejection step .new trajectories can be proposed in a variety of ways .one approach is to employ open - ended trajectory algorithms , which generate new trajectories by directly integrating the dynamics of the system .examples of this approach include the widely used `` shooting '' and `` shifting '' techniques. an alternative approach employs double - ended trajectory algorithms , in which the dynamical trajectories are treated as extended geometrical objects , or chains - of - states, and sampled using path integral monte carlo methods .  in practice ,two factors have contributed to the predominance of tps applications that employ opened - ended trajectories. firstly , for processes that react by way of a single barrier - crossing event , the open - ended shooting and shifting algorithms are very effective in providing decorrelated members of the transition path ensemble , thus rendering a double - ended approach unnecessary .secondly , the path integral sampling problem that arises in a double - ended formulation of tps has appeared to be prohibitively difficult .however , strong motivation for the development of double - ended tps algorithms does exist .many complex systems , including many biologically relevant systems , undergo transition processes that involve slow , diffusive barrier crossing dynamics and exhibit the possibility of metastable intermediates .these conditions are known to cause difficulty for the open - ended tps algorithms, but they are naturally overcome in double - ended methods which enable the trajectory endpoint positions to be constrained to the reactant and product regions .furthermore , recent developments in path integral monte carlo methods and the availability of massively parallel computers suggests that a double - ended formulation of tps can be made tractable .  in the present work ,we consider the issue of sampling double - ended trajectories for diffusive dynamics .going beyond the coarse - graining efforts of elber and coworkers, we seek the development of an efficient , ergodic sampler of the transition path ensemble .we show in section ii that the transition path ensemble for diffusive dynamics can be expressed using a _symmetric onsager - machlup formula_. in section iii , we show that discretization of this formula gives rise to a symmetric numerical integrator that requires only first derivatives of the potential energy function and exhibits second order accuracy .we draw contrast with the standard onsager - machlup formula, the discretization of which gives rise to the stochastic euler integrator that is only first order in accuracy . in section iv, we address the path integral monte carlo sampling problem that accompanies a double - ended tps approach .we begin by establishing the need for fast sampling the fine features of the stochastic process .we also argue that additional methods are needed to efficiently sample the coarse features of trajectories that are long in time .we then introduce the _ sliding and sampling _ algorithm that alleviates this coarse - feature sampling problem and gives rise to a natural parallelization scheme . in section v ,we utilize the s&s algorithm to sample the ensemble of reactive trajectories for the low - temperature interconversion of the 38-atom lennard - jones cluster between its octahedral and icosahedral minima .the time - evolution of relatively large particles in a highly viscous thermal bath is described by overdamped stochastic dynamics . in one dimension ,these dynamics , which are also known as brownian or diffusive dynamics , are governed by the law here , and the particle position and velocity at time , and is the external potential energy function .the quantity is the friction coefficient , which is related to the diffusion coefficient ( ) via the reciprocal temperature .the white noise is formally expressed as the time derivative of a wiener process , or brownian motion , of variance .extension of eq .( [ eq : diffusiveeom ] ) , and the rest of this presentation , to higher dimensions is trivial .  the object of primary interest in this study is the distribution of transition paths, where is the equilibrium distribution of trajectories that obey eq .( [ eq : diffusiveeom ] ) and pass through points in subsequent timesteps of . is the indicator function for the subset of configuration space associate with the reactant species , and is the corresponding product indicator function . from the markov property of diffusive dynamics , it follows that in the canonical ensemble where is the conditional probability that the particle arrives at point , given that it started at point some time earlier . obeys the smoluchowski equation , (x',x;t)\right\},\ ] ] which is the partial differential equation that corresponds to eq .( [ eq : diffusiveeom ] ) . however , by reweighting the conditional probability , /2 } k(x',x;t),\ ] ] eq .( [ eq : smolu ] ) is cast in the self adjoint form of the bloch equation , (x',x;t),\ ] ] with .\ ] ] the solution for the conditional probability is then immediately known from the feynman - kac solution for the green s function of the bloch equation, with the symbol meaning an average over all brownian paths of variance that start at and end up at in time .  in practice , it will be useful to eliminate the evaluation of second derivatives of the potential function ( eq . ( [ eq : fkeq ] ) ) .we thus perform a final manipulation .using that we consider the difference between forward and backward ito integrals , obtaining  (w_{t_{k+1 } } - w_{t_k } ) \\ = \lim_{\delta t \to 0 } \sum_{k = 0}^{n-1 }  v '' ( w_{t_k})(w_{t_{k+1 } } - w_{t_k})^2 = 2d\lim_{\delta t \to 0 } \sum_{k = 0}^{n-1 }  v '' ( w_{t_k})\delta{t_k } = 2d\int_0^t v''(w_u)du.\end{aligned}\ ] ]  substituting the last equality in the eq .( [ eq : fkeq ] ) , we finally obtain the _ symmetric onsager - machlup _formula   the symmetric onsager - machlup formula in eq . ([ eq : som ] ) , which to our knowledge is an original contribution of this work , is so called because of its clear relationship to the standard onsager - machlup formula, however , unlike eq .( [ eq : om ] ) , eq .( [ eq : som ] ) is symmetric in and .( that eq .( [ eq : som ] ) is symmetric is obvious from the inspection of eq .( [ eq : itomanip ] ) ; that eq .( [ eq : om ] ) is not symmetric follows from its dependence on a forward ito integral and the lack of equality of forward and backward ito integrals . )although the two solutions are formally equivalent , it will be shown in the next section that the symmetric formula gives rise to more stable numerical implementation .  before proceeding, we note that the feynman - kac equation is also symmetric in and and promises good numerical stability , but we will currently avoid evaluating second derivatives of the potential energy function and reserve exploration of this avenue for a later date .discretization of the standard onsager - machlup formula gives rise to euler s integrator, a first - order integration scheme . in this section, we will show that discretization of the symmetric onsager - machlup formula gives rise to a more stable , second - order integration scheme .a symmetric discretization can be obtained from the symmetric onsager - machlup formula by utilizing the finite difference appearing in eq .( [ eq : itomanip ] ) for the difference of ito integrals and the trapezoidal quadrature technique for the riemann integral .thus , letting for all , for all , and , we obtain the approximation (w_{t_k } - w_{t_{k-1 } } ) \right\}. \end{aligned}\ ] ] since the joint distribution of the wiener process is known explicitly , the right - hand side can be reduced to a finite - dimensional integral as follows .define the quantity and let then , setting and , (x_k - x_{k-1 } ) \right\}. \nonumber\end{aligned}\ ] ]  finally , the right - hand side of eq .( [ eq : i27 ] ) can be expressed in a way that is more familiar to those working with imaginary - time path integrals . has the form of a lie - trotter product obtained from the `` short - time '' approximation ( remember that ) (x ' - x ) \right\}. \qquad\end{aligned}\ ] ] eqs .( [ eq : tpsdist ] ) , ( [ eq : markov ] ) , ( [ eq : reweight ] ) , and ( [ eq : i29 ] ) fully specify the discretized distribution of transition paths employed in this study , i_b(x_n)e^{-\frac{\beta}{2 } v(x_n)}.\ ] ]  in considering the symmetric formula , we have been motivated not only by the need to ensure detailed balance , but also by the observation of suzuki that the order of convergence of a short - time approximation of the form ( which is generally one ) is improved when the symmetric form is utilized ( to the value of two , which is also the maximal value for such products ; a summary of suzuki s findings can be found in ref . ) . a more general theory , which is also valid outside the domain of applicability of the trotter theorem , has been designed by one of us. that theory can be utilized both to verify the convergence order of a short - time approximation as well as to design improved ones. it also allows one to design optimal propagation techniques useful for grid implementations.  we shall utilize such a grid technique to verify numerically that the convergence order of the short - time approximation given by eq .( [ eq : i29 ] ) is two .the grid technique is a sparse version of numerical matrix multiplication .the choice of grid and propagation technique depends on the order of convergence that is desired .for example , if the convergence order of a spatially - continuous short - time approximation is two , then a grid technique of order two preserves the order of convergence , whereas a grid technique of order greater or equal to three preserves not only the order but also the convergence constant . we shall therefore utilize the fourth - order grid from ref .  to verify the convergence order . once the order is verified and found not to exceed two , one can utilize the second order grid technique for actual computations on a grid . the lower the order of the grid is , the better the sparsity and the speed of the propagation scheme .grid techniques are generally limited to low dimensional systems .the physical system considered is taken from ref . .it is a two - dimensional two - channel problem described by the potential in reduced units , the friction coefficient is , the inverse temperature is , whereas the total propagation time is set to , twice the value utilized in the original reference .we shall verify the ability of the short - time approximation to preserve the boltzmann distribution .using eq .( [ eq : i21 ] ) , if we let and integrate over all coordinates , we should ideally obtain the value this is the case for the exact green s function , which leaves the distribution invariant .if the short - time approximation is utilized instead , then the result after propagation , denoted here by , is normally different from .it converges to in the limit .we want to establish how fast this convergence is .an easy way is to verify that the difference  ] .the widths of the proposal distributions are restricted to be the same for all path variables within a given -indexed layer , and the numerical values for these widths are tuned in an equilibration phase to yield approximately acceptance ..[tab : disp ] proposal distribution widths for the metropolis monte carlo moves in the two - dimensional example. [ cols="^,^,^,^",options="header " , ]   +  is the layer index for the lc representation of the fragment , is the number of time - slices moved by changing the path variable , is the equilibrated path variable distribution width , and is the corresponding distance in coordinate space .  in table [ tab : displj ] , we report the equilibrated path variable proposal widths for the simulation .the results are nearly identical to those obtained for , and they have the same features as were discussed for the two - dimensional example in connection with table [ tab : disp ] . in particular , we note a plateau in the values for as the path variables become decoupled from the potential energy surface .the plateau value seen here is smaller than was found in table [ tab : disp ] because we are now dealing with three - dimensional , rather than two - dimensional , particles . for , the decrease with decreasing while the corresponding values remain large , indicating that these layers remain productive in sampling the path .finally , for , the rapidly decrease as the most coarse layers approach the onset of stiffness .this table suggests that a fragmentation length of at least is necessary to avoid the exclusion of lc path variables that substantially contribute to the sampling efficiency .  for the purposes of analysis ,the sampled trajectories were quenched on the potential energy surface using the zero temperature string ( zts ) method of e , ren , and vanden - eijnden. given an initial path , this method yields a continuous , minimum - energy path ( i.e. one that is everywhere tangent to the gradient of the potential energy function and is compose of equidistant images of the system ) .we employed the zts method using our sampled trajectories as input .specifically , we linearly interpolated between every other time - slice in the sampled diffusive trajectory and performed the zts minimization using a path composed of images of the system .the zts calculations were performed in the absence of the cluster constraining potential , .convergence was determined to be reached when the `` minmax '' image , namely the highest energy image along the quenched path , exhibited an rms force of less than per atom .the potential energy profile for a typical transition path that has been quenched in this fashion , and which illustrates the ruggedness of lj potential energy suface , is shown in figure [ afterquench ] .   a typical quenched pathway for interconversion between the octahedral and icosahedral lj isomers , obtained by sampling with the s&s algorithm and quenching using the zts method ., width=321 ]  in fig .[ minmax ] , we present the time series of minmax values obtained from the two s&s simulations at reported here .the red and blue lines indicate the results for the and simulations , respectively .trajectories recorded during the equilibration stage of the simulation are included .the figure also includes two results from a discrete path sampling ( dps ) calculation provided by wales. the solid black line indicates the lowest minmax value found to connect the permutational isomers , an energy of .the dashed line indicates the minmax value of for the most probable path at the lower temperature of , according to the harmonic rate theory employed in the dps calculation .minmax values for lj interconversion trajectories sampled at using the s&s algorithm with fragment lengths =128 ( red ) and ( blue ) .trajectories are output every monte carlo sweeps .the solid and dashed black lines indicate the minmax values for the lowest energy path and most probable path at t=0.02 , respectively , according to a dps calculation by wales. the dps most probable path is determined by a harmonic rate theory. , width=321 ]  it is clear that the s&s algorithm at least partially succeeds in sampling the distribution of paths .it would not at all have been surprising if the sampler had simply gotten trapped in the region of high - energy pathways around the initial straight - line trajectory . instead , within the first output trajectories, it finds its way to lower minmax values and proceeds to explore many different transition pathways .it is especially encouraging that , upon finding a particular low - energy pathway , the sampler does not become endlessly trapped . in many instances , we see that a low minmax value is found , but then abandoned soon after in favor of higher values .this observation also suggests that finite temperature , and perhaps the finite trajectory time , play a role in biasing the ensemble of trajectories away from the global minimum energy path .the key role of temperature , and thus entropy , in shaping the transition path ensemble is corroborated by the dps calculations . even at the exceedingly low temperature of , the lowest energy transition path ( solid line )is not predicted to be the most favorable ( dashed line ) .it is therefore not surprising that our s&s simulations , which sample trajectory space at the phase change temperature of , prefer transition paths that are still higher in minmax energy .both s&s simulations do locate the pathway indicated by the dashed line in the figure , but they quickly move on in favor of more energetic paths .these observations highlight the importance of methods that actually sample trajectory space .  despite these encouraging observations, we can not claim that the s&s algorithm achieves fully ergodic sampling in either simulation . in both s&s simulations reported in fig .[ minmax ] , we see extended periods in which the sampler , although not necessarily locally trapped , appears to be confined within a `` tube '' of accessible trajectories .for example , beyond the first 100 output trajectories in the simulation , we see that the system consistently returns to a minmax value of . a similar recurrence in the simulationis seen at the minmax value of .these apparent transition tubes are analogous to the basins of stability found in configuration space , and the sampler s task of equilibrating them is difficult for the same reasons .further characterization of these transition tubes , perhaps using the finite temperature string method in coarse variables, is an interesting possibility for future work .sweep for sweep , the simulation shown in the blue should do a better job of sampling than the simulation shown in the red .the former employs twice the fragment length and thus attempts all of the same monte carlo moves , plus those associated with its most coarse lc layer . that the two simulations do not samplethe same transition tube is an indication that the , at least , is not ergodic , and most likely neither is .however , given the extreme ruggedness of the lj potential energy surface and the complexity of its low - temperature interconversion process , it is very encouraging to find that reasonable transition paths are obtained using the s&s algorithm and that the local sampling of path space , at least , is achieved .in this work , we have described several developments that aide the efficient transition path sampling of double - ended diffusive trajectories . in particular , we have introduced the symmetric onsager - machlup formula , which can be utilized as the basis for deriving high - order symmetric integrators that utilize only the forcefield , and we have shown that a direct time discretization on a uniform grid generates an approximation that is already accurate to second order . furthermore , we have shown that the sliding and sampling algorithm can naturally leverage massively parallel computer architectures to simulate double - ended trajectories on long timescales .  using a two - dimensional example ,we demonstrate the computational and wall - clock efficiency of the s&s algorithm .we find that , because of the layered ( multi - scale ) structure of the lc representation and the fact that the sliding moves eliminate the intrinsic asymmetries of this representation , the computational efficiency of the s&s algorithm is relatively insensitive to the length of fragments employed .this observation greatly simplifies the implementation of the algorithm - the fragment lengths can be primarily chosen according to the number of available nodes and the desire to minimize the wall - clock time of simulation .we also employ the s&s algorithm to study the low - temperature structural interconversion of the lj cluster .we find that , for the simulation times employed here , the s&s algorithm did not ergodically sample the full trajectory space - a result that is not at all surprising given the difficulty of equilibrating the low - temperature thermodynamics of this system with the direct metropolis algorithm .however , we are encouraged by finding that local sampling of path space is achieved with the s&s algorithm and that the sampler was able to find a large number of low - energy transition pathways .biological systems at room temperature do not exhibit the same free energy ruggedness that is found in the lj cluster .but they can still have metastable intermediates , entropic bottlenecks , and exceedingly long timescales , which frustrate open - ended trajectory sampling algorithms .the success of the direct metropolis monte carlo algorithm in sampling the configuration space of such systems is a good indication that the s&s algorithm will be similarly successful in sampling their trajectory space .it is in this regime of biologically relevant processes that we expect the s&s algorithm to find its most immediately applicability .this work was supported in part by the national science foundation grant no .che-0345280 and by the director , office of science , office of basic energy sciences , chemical sciences , geosciences , and biosciences division , u.s .department of energy under contract no .de - ac02 - 05ch11231 .the authors thank d. chandler , j. d. doll , e. vanden - 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Abstract: we address the problem of sampling double - ended diffusive paths . the ensemble of paths is expressed using a symmetric version of the onsager - machlup formula , which only requires evaluation of the force field and which , upon direct time discretization , gives rise to a symmetric integrator that is accurate to second order . efficiently sampling this ensemble requires avoiding the well - known stiffness problem associated with sampling infinitesimal brownian increments of the path , as well as a different type of stiffness associated with sampling the coarse features of long paths . the fine - feature sampling stiffness is eliminated with the use of the fast sampling algorithm ( fsa ) , and the coarse - feature sampling stiffness is avoided by introducing the sliding and sampling ( s&s ) algorithm . a key feature of the s&s algorithm is that it enables massively parallel computers to sample diffusive trajectories that are long in time . we use the algorithm to sample the transition path ensemble for the structural interconversion of the 38-atom lennard - jones cluster at low temperature .