Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

Randomly coloring random graphs

We consider the problem of generating a coloring of the random graph ??_n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β(p) such that the distribution is close to uniform in O(n log n) steps of the chain. This problem has been previously studied for ??_n,p in cases where np is relatively small. Here we consider the “dense” cases, where np ε [ω ln n, n] and ω = ω(n) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β(p) than can be achieved using more general techniques. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

A random CLT for dependent random variables

We introduce a new condition for {Y_τn} to have the same asymptotic distribution that {Y_n} has, where {Y_n} is a sequence of random elements of a metric space (S, d) and {τ_n} is a sequence of random indices. The condition on {Y_n} is that max_iDnd(Y_i, Y_an)→^p0 as n → ∞, where D_n = {i: |k_i−k_an| ≤ δ_ank_an} and {δ_n} is a nonincreasing sequence of positive numbers. The condition on {τ_n} is that P(|(k_τn/k_an)−1| > δ_an) → 0 as n → ∞. Under these conditions, we will show that d(Y_τn, Y_an) → ^P0 and apply this result to the CLT for a general class of sequences of dependent random variables.     

Delaying satisfiability for random 2SAT

Let (C₁,C′_(*)),(C₂,C′_(*)),…,(C _m,C′_(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 \begin{align*}\binom{n}{2}\end{align*} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on‐line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on‐line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^1/4. This contrasts with the well‐known fact that a random m ‐clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on‐line setting above, we also consider an off‐line version in which all m clause‐pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off‐line setting, we show that the two‐choice satisfiability threshold for k ‐SAT for any fixed k coincides with the standard satisfiability threshold for random 2k ‐SAT.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

A self normalized law of the iterated logarithm for random walk in random scenery

LetX,X _i ,i1, be a sequence of i.i.d. random vectors in ^d . LetS _o=0 and, forn1, letS _n =X ₁+...+X _n . LetY,Y(), ^d , be i.i.d. -valued random variables which are independent of theX _i . LetZ _n =Y(S _o )+...+Y(S _n ). We will callZ _n arandom walk in random scenery.In this work, we consider the law of the iterated logarithm for random walk in random sceneries. Under fairly general conditions, we obtain arandomly normalized law of the iterated logarithm.Supported in part by NSF Grants DMS-85-21586 and DMS-90-24961.     

The Reverse H‐free Process for Strictly 2‐Balanced Graphs

Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2‐balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs.     

Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

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     

A functional limit theorem for random graphs with applications to subgraph count statistics

We consider a random graph that evolves in time by adding new edges at random times (different edges being added at independent and identically distributed times). A functional limit theorem is proved for a class of statistics of the random graph, considered as stochastic processes. the proof is based on a martingale convergence theorem. the evolving random graph allows us to study both the random graph model K_{n, p}, by fixing attention to a fixed time, and the model K_{n, N}, by studying it at the random time it contains exactly N edges. in particular, we obtain the asymptotic distribution as n → ∞ of the number of subgraphs isomorphic to a given graph G, both for K_{n, p} (p fixed) and K_{n, N} (N/(ⁿ₂)→ p). the results are strikingly different; both models yield asymptotically normal distributions, but the variances grow as different powers of n (the variance grows slower for K_n, N; the powers of n usually differ by 1, but sometimes by 3). We also study the number of induced subgraphs of a given type and obtain similar, but more complicated, results. in some exceptional cases, the limit distribution is not normal.     

Critical random graphs and the structure of a minimum spanning tree

We consider the complete graph on n vertices whose edges are weighted by independent and identically distributed edge weights and build the associated minimum weight spanning tree. We show that if the random weights are all distinct, then the expected diameter of such a tree is Θ(n^1/3). This settles a question of Frieze and Mc‐Diarmid (Random Struct Algorithm 10 (1997), 5–42). The proofs are based on a precise analysis of the behavior of random graphs around the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erd?s‐Rényi random graph G(n,p) equals the number of nonisolated vertices for any c ln n/n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Dirac's theorem for random graphs

A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least \begin{align*}\left\lceil n/2 \right\rceil\end{align*} is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of random graph properties we prove that if p ? log n/n, then a.a.s. every subgraph of G(n,p) with minimum degree at least (1/2 + o (1) )np is Hamiltonian. Our result improves on previously known bounds, and answers an open problem of Sudakov and Vu. Both, the range of edge probability p and the value of the constant 1/2 are asymptotically best possible. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Explicit Bounds for the Return Probability of Simple Random Walks

We give an exact computation of the second order term in the asymptotic expansion of the return probability, P_2n^d(0,0), of a simple random walk on the d-dimensional cubic lattice. We also give an explicit bound on the remainder. In particular, we show that P_2n^d(0,0) < 2 (d/4n)^d/2 where n M=M(d) is explicitly given.     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

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     

Asymptotic behavior of random heaps

We consider a random walk W_n on the locally free group (or equivalently a signed random heap) with m generators subject to periodic boundary conditions. Let #T(W_n) denote the number of removable elements, which determines the heap's growth rate. We prove that lim_n→∞??(#T(W_n))/m ≤ 0.32893 for m ≥ 4. This result disproves a conjecture (due to Vershik, Nechaev and Bikbov [Comm Math Phys 212 (2000), 469–501]) that the limit tends to 1/3 as m → ∞. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

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.     

Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_n,p (above the connectivity threshold) and for random regular graphs G_r,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well‐known ErdJW‐RSAT1100590x.png ‐Renyi phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t^* all components are of size O(log n). For G_n,p and G_r we give the value of t^*, and the size of Γ(t). For G_r, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(log n). We also show that for random digraphs D_n,p above the strong connectivity threshold, there is a similar directed phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t^* all strongly connected components are of size O(log n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

The diameter of sparse random graphs

We derive an expression of the form c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence. The result holds asymptotically almost surely, assuming that certain convergence and supercriticality conditions are met, and is applicable to the classical random graph G_n,p with np = Θ(1) + 1, as well as certain random power law graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

A Grain of Dust Falling Through a Sierpinski Gasket

In this paper we analyze the downward random motion of a particle in a vertical, bounded, Sierpinski gasket G, where at each layer either absorption or delays are considered. In the case of motion with absorption the explicit distribution of the position of the descending particle in the pre-gasket Gn is obtained and the limiting case of the Sierpinski gasket discussed. For the delayed downward motion we derive a representation of the random time needed to arrive at the base of Gn in terms of independent binomial random variables （containing the contribution of delays at different layers with different geometrical structures）.