We study asymptotic behaviors of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by the Bernoulli variables. We first establish the existence and continuity including the negative infinity value of the coupling constant β. Our proof of existence at β=−∞ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at β=−∞ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.
The main part of Theorem is the continuity of @math around @math , which is the zero temperature asymptotic result for the free energy. This type of problems does not seem to attract much interest in the discrete time setting since in some cases the answers are simple. For instance, consider the (nearest-neighbor) simple random walk model with an i.i.d. random environment with @math . Then it is easy to see that as @math , the free energy is asymptotic to @math times the time constant of the directed last passage percolation. However, if @math is Bernoulli distributed and we send @math , the situation is not so simple. As we mentioned at the beginning, the existence of @math proved in @cite_19 is already highly nontrivial and the continuity as @math remains an open question at the moment.
@cite_19
1508206940
We study compositions of the integer @math in which the first part, successive differences, and the last part are constrained to lie in prescribed sets @math , respectively. A simple condition on @math guarantees that the generating function @math has only a simple pole on its circle of convergence and this at @math , a function independent of @math and @math . Thus the number of compositions is asymptotic to @math for a suitable constant @math . We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of @math , such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If @math has only finitely many positive elements @math , or finitely many negative elements @math , then the largest part and number of distinct part sizes are almost surely @math . On the other hand, when both @math and @math have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely @math , and we give sufficient conditions for the largest part to be almost surely asymptotic to @math . Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on @math with i.i.d. exponential weights on the vertices. Fix two points @math and @math for some @math , and consider the maximal paths @math and @math starting at @math and @math respectively to the point @math for @math . Our object of study is the point of coalescence, i.e., the point @math with smallest @math . We establish that the distance to coalescence @math scales as @math , by showing the upper tail bound @math for some @math . We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction @math starting from @math and @math , we establish the optimal tail estimate @math , for the point of coalescence @math . This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound. We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random @math -regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for @math the cover time is asymptotic to @math , where @math . In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph @math . For @math independent walks on @math , the cover time @math is asymptotic to @math , where @math is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is @math . If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for @math walks to broadcast a single piece of information to each other is asymptotic to @math as @math . We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of @math explosive particles ( @math even) tends to @math as @math . The case of @math coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to @math . Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with @math predators and @math prey the expected detection time is @math , where @math is the @math th harmonic number. An approximate Spielman-Teng theorem for the least singular value @math of a random @math square matrix @math is a statement of the following form: there exist constants @math such that for all @math , @math . The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for @math -valued matrices, each of whose rows is an independent vector with exactly @math zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a truly combinatorial' setting. We introduce a Gibbs measure on nearest-neighbour paths of length @math in the Euclidean @math -dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature @math . We prove that, for all @math , the random walk condensates to a set of diameter @math in dimension @math , up to a multiplicative constant. In all dimensions @math , we also prove that the volume is bounded above by @math and the diameter is bounded below by @math . Similar results hold for a random walk conditioned to have local time greater than @math everywhere in its range when @math is larger than some explicit constant, which in dimension two is the logarithm of the connective constant. It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In @math steps, the mean of the record distribution grows as the @math while the standard deviation grows as @math , so the distribution is non-self-averaging. The mean shortest and longest duration records grow as @math and @math , respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found. We consider the simple exclusion process with @math particles on a segment of length @math performing random walks with transition @math to the right and @math to the left. We focus on the case where the asymmetry in the jump rates @math vanishes in the limit when @math and @math tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than @math . We show that in the former case ( @math ), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e. constant @math ) case studied in [LL18], while the latter case ( @math ) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in [Lac16b]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order @math . The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as @math , etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical @math -Coloring . A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the @math -level Hierarchical @math -Coloring problem is @math , for @math . The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized @math -time algorithm solving the LCL can be transformed into a deterministic @math -time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges @math --- @math or @math --- @math . 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in @math time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be @math and @math . Given a finite graph @math , a vertex of the lamplighter graph @math consists of a zero-one labeling of the vertices of @math , and a marked vertex of @math . For transitive @math we show that, up to constants, the relaxation time for simple random walk in @math is the maximal hitting time for simple random walk in @math , while the mixing time in total variation on @math is the expected cover time on @math . The mixing time in the uniform metric on @math admits a sharp threshold, and equals @math multiplied by the relaxation time on @math , up to a factor of @math . For @math , the lamplighter group over the discrete two dimensional torus, the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . For @math when @math , the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . In particular, these three quantities are of different orders of magnitude. Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process. We consider a continuous time random walk on the rooted binary tree of depth @math with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by @math and then centered by @math , the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree. We study the asymptotic behaviour of the nodal length of random @math -spherical harmonics @math of high degree @math , i.e. the length of their zero set @math . It is found that the nodal lengths are asymptotically equivalent, in the @math -sense, to the "sample trispectrum", i.e., the integral of @math , the fourth-order Hermite polynomial of the values of @math . A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit. Let @math be constant, and let @math denote the set of r-regular graphs with vertex set V = 1,2,...,n . Let G be chosen randomly from @math . We prove that with high probability ( ) the cover time of a random walk on G is asymptotic to @math . In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph. Let @math be an undirected and connected graph with @math vertices and @math edges. The coalescence time, @math , is the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time for general connected graphs and prove that @math . Here @math is the second eigenvalue of the transition matrix of the random walk. To avoid problems arising from, e.g., lack of coalescence on bipartite graphs, we assume the random walk can be made lazy if required. The value of @math is given by @math , where @math is the degree of vertex @math , and @math is the average degree. The parame... In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any @math and for any exponent @math giving the rate of decay of the percolation process, we show that the return probability decays like @math up to logarithmic corrections, where @math denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest, the main result is needed for a companion paper studying the scaling limit of simple random walk on the infinite cluster. The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their @math -bit input strings is large (i.e., at least @math ) or small (i.e., at most @math ); they do not care if it is neither large nor small. This @math gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an @math lower bound on this problem was known only in the one-way setting. We prove an @math lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that @math -approximately counting the number of distinct elements in a data stream requires @math space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight @math lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an @math lower bound on the one-way randomized communication complexity. A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation @math added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, @math ) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous, since @math is found to scale as @math with @math . Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure. Let @math denote the first passage time to 1 of a standard Brownian motion. It is well known that as @math goes to infinity, @math goes to zero at rate @math , where @math equals @math . The goal of this note is to establish a quantitative, infinite dimensional version of this result. Namely, we will prove the existence of positive and finite constants @math and @math , such that for all @math , @math where @math ' denotes the natural logarithm, and @math is the Fukushima-Malliavin capacity on the space of continuous functions. We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math .