We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph @math is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on @math , scaled by the number of edges in @math . This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was @math for @math -vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000).
Furthermore, for a few families of specific examples, the asymptotics of the cover time have been calculated more precisely. These include the work of Aldous @cite_37 for regular trees, Dembo, Peres, Rosen, and Zeitouni @cite_22 for the 2-dimensional discrete torus, and Cooper and Frieze @cite_45 for the giant component of various random graphs.
@cite_37 @cite_45 @cite_22
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We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length @math . Dembo, Peres, Rosen, and Zeitouni [Ann. Math. 160:433-464, 2004] identified the leading term in the asymptotics for the cover time as @math goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler [Probab. Theory Relat Fields. 167:461-552, 2017]. Abstract For simple random walk on a finite tree, the cover time is the time taken to visit every vertex. For the balanced b-ary tree of height m, the cover time is shown to be asymptotic to 2m 2 b m + 1 ( log b) (b − 1) as m → ∞ . On the uniform random labeled tree on n vertices, we give a convincing heuristic argument that the mean time to cover and return to the root is asymptotic to 6(2π) 1 2 n 3 2 , and prove a weak O(n 3 2 ) upper bound. The argument rests upon a recursive formula for cover time of trees generated by a simple branching process. The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Renyi random graph G(n, c n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Renyi random graph G(n, 1 n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube 0, 1 n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3 2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4. In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time. Aldous and Pitman have studied the asymptotic behavior of the additive coalescent processes using a nested family random forests derived by logging certain inhomogeneous continuum random trees. Here we propose a different approach based on partitions of the unit interval induced by certain bridges with exchangeable increments. The analysis is made simple by an interpretation in terms of an aggregating server system. This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in Aldous and Fill (Reversible Markov chains and random walks on graphs, in preparation). We also derive some corollaries which qualitatively describe “how” covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman (Ann Math (2) 171(3):2039–2087, 2010), and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest. Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-Lof, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph G(n,p) above the phase transition. Here we show that the same method applies to the analogous model of random k -uniform hypergraphs, establishing asymptotic normality throughout the (sparse) supercritical regime. Previously, asymptotic normality was known only towards the two ends of this regime. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc. In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494…)n-many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximately @math n, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 Preliminary version of this paper appeared in Proceedings of the Seventeenth ACM-SIAM Symposium on Discrete Algorithms 2006. ArXive version. The theme of this work is an "inside-out" approach to the enumeration of graphs. It is based on a well-known decomposition of a graph into its 2-core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number of 2-cores with a given number of vertices and edges, we solve the corresponding enumeration problem for the connected 2-cores. For a subrange of the parameters, we also enumerate those 2-cores by using a deeper inside-out notion of a kernel of a connected 2-core.Using this enumeration result in combination with Caley's formula for forests, we obtain an alternative and simpler proof of the asymptotic formula of Bender, Canfield and McKay for the number of connected graphs with n vertices and m edges, with improved error estimate for a range of m values.As another application, we study the limit joint distribution of three parameters of the giant component of a random graph with n vertices in the supercritical phase, when the difference between average vertex degree and 1 far exceeds n-1 3. The three parameters are defined in terms of the 2-core of the giant component, i.e. its largest subgraph of minimum degree 2 or more. They are the number of vertices in the 2-core, the excess (#edges - #vertices) of the 2-core, and the number of vertices not in the 2-core. We show that the limit distribution is jointly Gaussian throughout the whole supercritical phase. In particular, for the first time, the 2-core size is shown to be asymptotically normal, in the widest possible range of the average vertex degree. LetT (x;") denote the rst hitting time of the disc of radius " centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x2T2T (x;")=j log"j 2 ! 2= as " ! 0. The same applies to Brownian motion on any smooth, compact connected, two- dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z 2 is asymptotic to 4n 2 (logn) 2 = . Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied non-rigorously in the physics literature. We also establish a conjecture, due to Kesten and R ev esz, that describes the asymptotics for the number of steps needed by simple random walk in Z 2 to cover the disc of radius n. 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 cover time of a random walk on graphs G e Gn,p when @math . We prove that whp, the cover time, is asymptotic to @math . ©Wiley Periodicals, Inc. Random Struct. Alg., 2007 We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds (P. Matthews, 1988) on the cover time, C, and prove that M 2<C= O(M(lnlnn) sup 2 ). We give a deterministic-polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((lnlnn) sup 2 ), improving the previous bound O(lnn) due to Matthews. The blanket time B was introduced by P. Winkler and D. Zuckerman (1996): it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C spl les B. Winkler and Zuckerman conjectured B=O(C) and proved B=O(Clnn). Our bounds above are also valid for the blanket time, and so it follows that B=O(C(lnlnn) sup 2 ). We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009). Author(s): Aldous, DJ; Pittel, B | Abstract: A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1 n, is studied. The detailed picture of emergence of giant components with O(n2 3) vertices is shown to be the same as in the Erdos-Renyi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t = π 2, rather than t = 1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum-of-squares and sum-of-cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky-type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812-854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley a Sons, Inc. Random Struct. Alg., 17, 79-102, 2000. Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime. We compute the second order correction for the cover time of the binary tree of depth n by (continuous-time) random walk, and show that with probability approaching 1 as n increases, τcov=|E|[2log2⋅n−logn 2log2+O((loglogn)8)], thus showing that the second order correction differs from the corresponding one for the maximum of the Gaussian free field on the tree. 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. We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus,and show that in both cases with probability approaching @math as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progresstowards a conjecture of Bramson and Zeitouni (2009). The ( )-cover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance ( >0 ) from any point. Its leading order in the small ( )-regime has been established by (Ann Math 160:433–464, 2004). In this work, the second order correction is identified. The approach relies on a multi-scale refinement of the second moment method, and draws on ideas from the study of the extremes of branching Brownian motion.