A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least αα, for some α∈(0,1)α∈(0,1). We also give general bounds on the total variation distance of a reversible chain at time tt in terms of the probability that some “worst” set of stationary measure at least αα was not hit by time tt. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to ∞∞.
This aforementioned connection was substantially refined recently by Peres and Sousi ( @cite_1 Theorem 1.1) and independently by Oliveira ( @cite_5 Theorem 2). Their approach relied on the theory of random times to stationarity combined with a certain de-randomization" argument which shows that for any lazy reversible irreducible finite chain and any stopping time @math such that @math , @math . As a (somewhat indirect) consequence, they showed that for any @math (this was extended to @math in @cite_19 ), there exist some constants @math such that for any lazy reversible irreducible finite chain
@cite_19 @cite_5 @cite_1
1593631888 2000931246 2139964991 2026924428
Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, @math . The sequence is said to exhibit @math -cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions. In this work we give a characterization of @math -cutoff for an arbitrary sequence of initial distributions @math (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. @math . Consider a Markov chain on @math whose stationary distribution in @math . Let @math be the expected hitting time of the worst set of size at least @math . It was recently proved by Peres and Sousi and independently by Oliveira that @math captures the order of the mixing time. In this work we further refine this connection and show that @math -cutoff can be characterized in terms of concentration of hitting times (starting from @math ) of sets which are worst in expectation w.r.t. @math . Conversely, we construct a counter-example which demonstrates that in general cutoff (as opposed to cutoff w.r.t. a certain sequence of initial distributions) cannot be characterized in this manner. Finally, we also prove that there exists an absolute constant @math such that for every Markov chain @math , for all @math , where @math is the inverse of the spectral gap of the chain. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We consider a finite random walk on a weighted graph G; we show that the fraction of time spent in a set of vertices A converges to the stationary probability @math with error probability exponentially small in the length of the random walk and the square of the size of the deviation from @math . The exponential bound is in terms of the expansion of G and improves previous results of [D. Aldous, Probab. Engrg. Inform. Sci., 1 (1987), pp. 33--46], [L. Lovasz and M. Simonovits, Random Structures Algorithms , 4 (1993), pp. 359--412], [M. Ajtai, J. Komlos, and E. Szemeredi, Deterministic simulation of logspace, in Proc. 19th ACM Symp. on Theory of Computing, 1987]. We show that taking the sample average from one trajectory gives a more efficient estimate of @math than the standard method of generating independent sample points from several trajectories. Using this more efficient sampling method, we improve the algorithms of Jerrum and Sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic Ising system, and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail.