On the fragmentation of a torus by random walk

We consider a simple random walk on a discrete torus \input amssym $({\Bbb Z}/N{\Bbb Z})^d$ with dimension d ≥ 3 and large side length N. For a fixed constant u ≥ 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the random walk in its first [uN^d] steps. We prove the existence of two distinct phases of the vacant set in the following sense: If u > 0 is chosen large enough, all components of the vacant set contain no more than (log N)^λ(u) vertices with high probability as N tends to infinity. On the other hand, for small u > 0, there exists a macroscopic component of the vacant set occupying a nondegenerate fraction of the total volume N^d. In dimensions d ≥ 5, we additionally prove that this macroscopic component is unique by showing that all other components have volumes of order at most (log N)^λ(u). Our results thus solve open problems posed by Benjamini and Sznitman, who studied the small u regime in high dimension. The proofs are based on a coupling of the random walk with random interlacements on \input amssym ${\Bbb Z}^d$ . Among other techniques, the construction of this coupling employs a refined use of discrete potential theory. By itself, this coupling strengthens a result by Windisch. © 2011 Wiley Periodicals, Inc.     

The VC dimension of k‐uniform random hypergraphs

A set of vertices is shattered in a hypergraph if any of its subsets is obtained as the intersection of an edge with the set. The VC dimension is the size of the largest shattered subset. Under the binomial model of k‐uniform random hypergraphs, the threshold function for the VC dimension to be larger than a given integer is obtained. The same is done for the testing dimension, which is the largest integer d such that all sets of cardinality d are shattered. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The type problem for random walks on trees

We consider the question of whether the simple random walk (SRW) on an infinite tree is transient or recurrent. For random-trees (all vertices of distancen from the root of the tree have degreed _n , where {d _n } are independent random variables), we prove that the SRW is a.s. transient if lim inf _n n E(log(d _n-1))>1 and a.s. recurrent if lim sup _n n E(log(d _n-1))<1. For random trees in which the degrees of the vertices are independently 2 or 3, with distribution depending on the distance from the root, a partial classification of type is obtained.Research supported in part by NSF DMS 8710027.     

Bounds on the cover time

Consider a particle that moves on a connected, undirected graphG withn vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. Tocover time is the first time when the particle has visited all the vertices in the graph starting from a given vertex. In this paper, we present upper and lower bounds that relate the expected cover time for a graph to the eigenvalues of the Markov chain that describes the random walk above. An interesting consequence is that regular expander graphs have expected cover time (n logn).This research was done while this author was a postdoctoral fellow in the Department of Computer Science, Princeton University, and it was supported in part by DNR grant N00014-87-K-0467.     

The degree sequence of a scale‐free random graph process

Recently, Barabási and Albert [2] suggested modeling complex real‐world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd^?γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n^1/15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

Critical window for the vacant set left by random walk on random regular graphs

We consider the simple random walk on a random d ‐regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time \begin{align*}\left\lfloor un \right\rfloor\end{align*}, where u > 0 is a fixed positive parameter. It was shown in ?erný et al., (Ann Inst Henri Poincaré Probab Stat 47 (2011) 929–968) that this so‐called vacant set exhibits a phase transition at u = u_?: there is a giant component if u < u_? and only small components when u > u_?. In this paper we show the existence of a critical window of size n^‐1/3 around u_?. In this window the size of the largest cluster is of order n^2/3. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues ν _j of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates ν _j . We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper’s final section we present extensions of our results to more general chains. Research supported by NSF grant DMS–0406104, and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Local Semicircle Law for Random Regular Graphs

We consider random d‐regular graphs on N vertices, with degree d at least (log N)⁴. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing (up to a logarithmic correction). Aside from well‐known consequences for the local eigenvalue distribution, this result implies the complete (isotropic) delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity.© 2017 Wiley Periodicals, Inc.     

Cycles with a given number of vertices from each partite set in regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d ⁺(x) and d ⁻(x) the outdegree and the indegree of x, respectively. A digraph D is called regular, if there is a number p ∈ ℕ such that d ⁺(x) = d ⁻(x) = p for all vertices x of D. A c-partite tournament is an orientation of a complete c-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether c-partite tournaments with r vertices in each partite set contain a cycle with exactly r − 1 vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if c = 2. If c ⩾ 3, then we will show that a regular c-partite tournament with r ⩾ 2 vertices in each partite set contains a cycle with exactly r − 1 vertices from each partite set, with the exception of the case that c = 4 and r = 2.     

The relaxed game chromatic number of outerplanar graphs

The (r,d)‐relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color α is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with α, and every neighbor of u that has already been colored with α is adjacent to at most d – 1 vertices that have already been colored with α. Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)‐relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)‐relaxed coloring game on G. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:69–78, 2004     

Some properties of random walk paths

For random walk on the d-dimensional integer lattice we consider again the problem of deciding when a set is recurrent, that is visited infinitely often with probability one by the random walk in question. Some special cases are considered, among them the following: for d = 2, what sequences (n_j) have the property that with probability one the random walk visits the origin for infinitely many n_j. A related problem, which is however not a special case of the recurrence problem, is to decide for what sequences (n_j) the states visited by the random walk at times n_j are all distinct, with only a finite number of exceptions. This problem is dealt with in the final part of the paper.     

Asymptotics for Euclidean minimal spanning trees on random points

Summary Asymptotic results for the Euclidean minimal spanning tree onn random vertices inR ^d can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process in allR ^d. In particular we prove a conjecture of Robert Bland: the sum of thed'th powers of the edge-lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofR ^d tends to a constant asn.Whether the limit forest is in fact a single tree is a hard open problem, relating to continuum percolation.Research supported by N.S.F. Grants MCS87-11426 and MCS 90-01710Research supported in part by N.S.F. Grant DMS88-12868, A.F.O.S.R. Grant 89-0301, ARO Grant DAAL03-89-G-0092 and NSA Grant MDA-904-H-2034     

Generating Random Regular Graphs

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n^1/3−ε), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd²). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs. * Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.     

Cycles through specified vertices

Recently, various authors have obtained results about the existence of long cycles in graphs with a given minimum degreed. We extend these results to the case where only some of the vertices are known to have degree at leastd, and we want to find a cycle through as many of these vertices as possible. IfG is a graph onn vertices andW is a set ofw vertices of degree at leastd, we prove that there is a cycle through at least vertices ofW. We also find the extremal graphs for this property.Research supported in part by NSF Grant DMS 8806097     

Higher-dimensional voronoi diagrams in linear expected time

A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram ofn independent and identically distributed points. The method is applied to derive exact asymptotic bounds on the expected number of vertices of the Voronoi diagram of points chosen from the uniform distribution on the interior of ad-dimensional ball; it is shown that in this case, the complexity of the diagram is ∵(n) for fixedd. An algorithm for constructing the Voronoid diagram is presented and analyzed. The algorithm is shown to require only ∵(n) time on average for random points from ad-ball assuming a real-RAM model of computation with a constant-time floor function. This algorithm is asymptotically faster than any previously known and optimal in the average-case sense. Based upon work supported by the National Science Foundation under Grant No. CCR-8658139 while the author was a student at Carnegie-Mellon University.     

A random tree model associated with random graphs

Grow a tree on n vertices by starting with no edges and successively adding an edge chosen uniformly from the set of possible edges whose addition would not create a cycle. This process is closely related to the classical random graph process. We describe the asymptotic structure of the tree, as seen locally from a given vertex. In particular, we give an explicit expression for the asymptotic degree distribution. Our results an be applied to study the random minimum-weight spanning tree question, when the edge-weight distribution is allowed to vary almost arbitrarily with n.     

Improved boolean formulas for the Ramsey graphs

In the theory of the random graphs, there are properties of graphs such that almost all graphs satisfy the property, but there is no effective way to find examples of such graphs. By the well-known results of Razborov, for some of these properties, e.g., some Ramsey property, there are Boolean formulas in ACC representing the graphs satisfying the property and having exponential number of vertices with respect to the number of variables of the formula. Razborov's proof is based on a probabilistic distribution on formulas of n variables of size approximately nd² log d, where d is a polynomial in n, and depth 3 in the basis { ∧, ⊕} with the following property: The restriction of the formula randomly chosen from the distribution to any subset A of the Boolean cube {0, 1}ⁿ of size at most d has almost uniform distribution on the functions A → {0, 1}. We show a modified probabilistic distribution on Boolean formulas which is defined on formulas of size at most nd log² d and has the same property of the restrictions to sets of size at most d as the original one. This allows us to obtain formulas the complexity of which is a polynomial of a smaller degree in n than in Razborov's paper while the represented graphs satisfy the same properties.     

Vertex degree of random geometric graph on exponentially distributed points

In this article, we consider exponential random geometric graph, in d-dimensional space (d≥2). The main results are almost-sure asymptotic rates of convergence/divergence for the maximum and minimum vertex degrees of graph, when edge distance varies with the number of vertices.     

The geometry of t‐spreads in k‐walk‐regular graphs

A graph is walk‐regular if the number of closed walks of length ? rooted at a given vertex is a constant through all the vertices for all ?. For a walk‐regular graph G with d+1 different eigenvalues and spectrally maximum diameter D=d, we study the geometry of its d‐spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three‐dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k‐walk‐regular graphs, a family which includes both walk‐regular and distance‐regular graphs, and their t‐spreads or vertices at distance t from each other. © 2009 Wiley Periodicals, Inc. J Graph Theory 64:312–322, 2010