Let $\mathcal{C}_1$ denote the largest connected component of the critical Erdős--Rényi random graph $G(n,{\frac{1}{n}})$. We show that, typically, the diameter of $\mathcal{C}_1$ is of order $n^{1/3}$ and the mixing time of the lazy simple random walk on $\mathcal{C}_1$ is of order $n$. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size $n^{2/3}$ of $p$-bond percolation on any $d$-regular $n$-vertex graph where such clusters exist, provided that $p(d-1)\le1+O(n^{-1/3})$.
Diameter and mixing time of the giant component in the percolated hypercube
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究院）；D-MATH ETH Zürich（苏黎世联邦理工学院数学系）；Institute of Statistics and Mathematical Methods in Economics, TU Wien（维也纳工业大学统计与经济数学方法研究所）；School of Computer Science, University of Sheffield（谢菲尔德大学计算机学院）
Let $g$, $h$ be a random pair of generators of $G=Sym(n)$ or $G=Alt(n)$. We show that, with probability tending to $1$ as $n\to \infty$, (a) the diameter of $G$ with respect to $S = \{g,h,g^{-1},h^{-1}\}$ is at most $O(n^2 (\log n)^c)$, and (b) the mixing time of $G$ with respect to $S$ is at most $O(n^3 (\log n)^c)$. (Both $c$ and the implied constants are absolute.) These bounds are far lower than the strongest worst-case bounds known (in Helfgott--Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap. Our r…
In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.
In this paper we study the diameter of Inhomogeneous random graphs $G(n,κ,p)$ that are induced by irreducible kernels $κ$. The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erdős-Rényi model $G(n,p)$ for several ranges of $p$. We find upper and lower bounds for the diameter of $G(n,κ,p)$ in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space.
Given a group G, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any $ε> 0$ and any family of groups G_k of order n_k for which $n_k \to \infty$, a graph $Γ_k \in \mathcal{G}(G_k,p)$ with high probability has diameter at most 2 if $p \geqslant \sqrt{(2 + ε) \frac{\log{n_k}}{n_k}}$ and with high probability has diameter greater than 2 if $p \leqslant \sqrt{(1/4 + ε)\frac{\log{n_k}}{n_k}}$. We also provide examples of families of gra…
This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions assoc…
The unit ball random geometric graph $G=G^d_p(λ,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most $λ$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity threshold: an asymptotic value for $λ$ in terms of $n$, above which $G$ is connected and below which $G$ is disconnected (and in fact has isolated vertices in most cases). In the connected zone, we determine upper and lower bounds for the graph diameter of $G$. Specifically, almost always, $\di…
We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, …
Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k). The same also holds when the generating set is taken to be a symmetric set of size 2k.
For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further…
In this paper we study the diameter of the random graph $G(n,p)$, i.e., the the largest finite distance between two vertices, for a wide range of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a simple proof of an essentially best possible result, with an $O_p(1)$ additive correction term. Using similar techniques, we establish 2-point concentration in the case that $np\to\infty$. For $p=(1+ε)/n$ with $ε\to 0$, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an $O_p(1/ε)$ additive correction term whose (approp…
Quantum walks on graphs have been shown in certain cases to mix quadratically faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. Here, we construct a lifted Markov chain on a graph with $n^2 D(G)$ vertices that mixes exactly to the average mixing distribution of a quantum walk on the graph $G$ with $n$ vertices, where $D(G)$ is the diameter of $G$. Moreover, the mixing time of this chain is $D(G)$ timesteps, and we prove …
We study the diameter of LPS Ramanujan graphs $X_{p,q}$. We show that the diameter of the bipartite Ramanujan graphs is greater than $ (4/3)\log_{p}(n) +O(1)$ where $n$ is the number of vertices of $X_{p,q}$. We also construct an infinite family of $(p+1)$-regular LPS Ramanujan graphs $X_{p,m}$ such that the diameter of these graphs is greater than or equal to $ \lfloor (4/3)\log_{p}(n) \rfloor$. On the other hand, for any $k$-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than $(1+ε)\log_{k-1}(n)$. We also have some numerical expe…
We study growth properties of the number of paths of lenght k for a variant of Cameo graphs introduced in an earlier paper. Sharp results are obtained for threshold for the k-path connectivity and the essential diameter.
We study the diameter of $C_1$, the largest component of the Erdős-Rényi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+ε}n$ where $ε^3 n \to \infty$ and $ε=o(1)$. This parameter was extensively studied for fixed $ε> 0$, yet results for $ε=o(1)$ outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when $ε^3 n\to\infty$ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, …
In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter ra…
We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in $\Z^d$. We show that for $d \geq 2$ and $p > p_c(\Z^d)$, the mixing time of simple random walk on the largest cluster inside $\{-n,...,n\}^d$ is $Θ(n^2)$ - thus the mixing time is robust up to constant factor.
Consider the random hypercube $H_2^n(p_n)$ obtained from the hypercube $H_2^n$ by deleting any given edge with probabilty $1-p_n$, independently of all the other edges. A diameter path in $H_2^n$ is a longest geodesic path in $H_2^n$. Consider the following two ways of tampering with the random graph $H_2^n(p_n)$: (i) choose a diameter path at random and adjoin all of its edges to $H_2^n(p_n)$; (ii) choose a diameter path at random from among those that start at $0=(0,..., 0)$, and adjoin all of its edges to $H_2^n(p_n)$. We study the question of whether these tamperings are detectable asympto…
In this paper we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most (ln n lnln n)^{1/2}, proving that the mixing time in this case is O((ln n/d)^2) asymptotically almost surely. As the average degree grows these become negligible and it is the diameter of the largest component that takes over, yielding mixing time O(ln n/ln d). We proved these results during the 2003-04 academic year. Similar results bu…
We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $n/2$. We show that the diameter of those random surfaces is asymptotic to $2 \log n$ in probability as $n \to \infty$.
We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.
We study random walks on the integers mod $G_n$ that are determined by an integer sequence $\{ G_n \}_{n \geq 1}$ generated by a linear recurrence relation. Fourier analysis provides explicit formulas to compute the eigenvalues of the transition matrices and we use this to bound the mixing time of the random walks.
We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends to infinity, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. The main result covers the case of uniformly distributed points within a $d$-dimensional ellipsoid with a unique major axis. Moreover, several generalizations of the main result are established, for example a limit l…
Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for $α> \tfrac{1}{2}$, $C\in\mathbb{R}$, $n\in\mathbb{N}$, set $R=2\ln n+C$ and build the graph $G=(V,E)$ with $|V|=n$ as follows: For each $v\in V$, generate i.i.d. polar coordinates $(r_{v},θ_{v})$ using the joint density function $f(r,θ)$, with $θ_{v}$ chosen uniformly from $[0,2π)$ and $r_{v}$ with density $f(r)=\frac{α\sinh(αr)}{\cosh(αR)-1}$ for $0\le…
We study in this paper, the first passage percolation on a random graph model, the configuration model. We first introduce, the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists of describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distributi…
The Diaconis--Gangolli random walk is an algorithm that generates an almost uniform random graph with prescribed degrees. In this paper, we study the mixing time of the Diaconis--Gangolli random walk restricted on $n\times n$ contingency tables over $\mathbb{Z}/q\mathbb{Z}$. We prove that the random walk exhibits cutoff at $\frac{n^2}{4(1- \cos{\frac{2 π}{q}})} \log n, $ when $\log q=o\left (\frac{\sqrt{\log n}}{\log \log n}\right )$.
We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. $d$-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max-$U$-statistic, hence a maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obta…
It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ local…
We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf_{n \to \infty} L_n/D_n =2$ and $\limsup_{n \to \infty} L_n /D_n = …
Bernoulli Institute, Groningen University（格罗宁根大学伯努利研究所）；Department of Ecology and Evolutionary Biology, Princeton University（普林斯顿大学生态与进化生物学系）；Utrecht Uniersity（乌得勒支大学）
Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $η_r>0$ such that $\text{diam}(D(n,r))=(1+η_r+o(1))\log_r{n}$. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $π_{\max}$ and $π_{\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $π_{\max} = n^{-1+o(1)}$ and $π_{\min}=n^{-(1…
It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2^k\ln 2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas, and inspect another parameter -- the diameter of the solution space (that is the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below th…
We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic factors).
We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are necessary and suffice for convergence to the stationary distribution.
We consider the generalised PageRank walk on a digraph $G$, with refresh probability $α$ and resampling distribution $λ$. We analyse convergence to stationarity when $G$ is a large sparse random digraph with given degree sequences, in the limit of vanishing $α$. We identify three scenarios: when $α$ is much smaller than the inverse of the mixing time of $G$ the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behaviour; when $α$ is much larger than the inverse of the mixing time of $G$ on the contrary one has pure exponential decay with rate $α$; when $α$ …
It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately $k^{-(τ-1)}$ with $τ\in(2,3)$, typical distances between pairs of vertices in a graph of size $n$ are asymptotic to $\frac{2\log\log n}{|\log(τ-2)|}$ and $\frac{4\log\log n}{|\log(τ-2)|}$, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order $\log\log n$ precisely when the minimal forward…
We consider the complete graph $\cK_n$ on $n$ vertices with exponential mean $n$ edge lengths. Writing $C_{ij}$ for the weight of the smallest-weight path between vertex $i,j\in [n]$, Janson showed that $\max_{i,j\in [n]} C_{ij}/\log{n}$ converges in probability to 3. We extend this result by showing that $\max_{i,j\in [n]} C_{ij} - 3\log{n}$ converges in distribution to a limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centered graph …
We prove that the spectral gap of a finite planar graph $X$ is bounded by $λ_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Curien on the mixing times of the simple random walk on planar graphs.
We show that the diameter of the directed configuration model with $n$ vertices rescaled by $\log n$ converges in probability to a constant. Our assumptions are the convergence of the in- and out-degree of a uniform random vertex in distribution, first and second moment. Our result extends previous results on the diameter of the model and applies to many other random directed graphs.
We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $\log g$ as $g \to \infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.
The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an $n$-dimensional polytope $P$ defined by the intersection of $m$ i.i.d.\ half-spaces whose normals are chosen uniformly from the sphere, we show that $\operatorname{diam}(P)$ is $Ω(n m^{\frac{1}{n-1}})$ and $O(…
We study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends on the product of the weights of its endpoints. We then study the minimum spanning tree corresponding to the edge capacities. Under a condition of finite moments on the node weights, we show that the expected diameter and typical distances of this minimum spanning tree are of order $n^{1/3}$. This is a generalization of the results of Addario-Berry, Broutin, …
We prove that every random walk in a uniformly elliptic random environment satisfying the cone mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.
We consider random walks in strong-mixing random Gibbsian environments in $\mathbb{Z}^d, d\ge 2$. Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha's conditional law of large numbers (CLLN) for mixing environment Rassoul-Agha (2005). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ($d\ge 5$).
Many regular graphs admit a natural partition of their edge set into cliques of the same order such that each vertex is contained in the same number of cliques. In this paper, we study the mixing rate of certain random walks on such graphs and we generalize previous results of Alon, Benjamini, Lubetzky and Sodin regarding the mixing rates of non-backtracking random walks on regular graphs.
We prove an upper bound of $1.5324 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling.
In this work we present different results concerning mixing properties of multivariate infinitely divisible (ID) stationary random fields. First, we derive some necessary and sufficient conditions for mixing of stationary ID multivariate random fields in terms of their spectral representation. Second, we prove that (linear combinations of independent) mixed moving average fields are mixing. Further, using a simple modification of the proofs of our results we are able to obtain weak mixing versions of our results. Finally, we prove the equivalence of ergodicity and weak mixing for multivariate …
We prove a law of large numbers for a class of multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments of some random times depending on the path. It applies in the non-nestling case, but we also provide examples of nestling walks that satisfy our assumptions. The derivation is based on an adaptation, using coupling, of the regeneration argument of Sznitman-Zerner.
We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non-trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d.\ environments.
