On the trace of random walks on random graphs

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random graph $G\sim G(n,p)$ for $p>C\ln{n}/n$ is, with high probability, Hamiltonian and $\Theta(\ln{n})$-connected. In the special case $p=1$ (i.e. when $G=K_n$), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the $k$'th time, the trace becomes $2k$-connected.Comment: 32 pages, revised versio

Vacant sets and vacant nets: Component structures induced by a random walk

Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$.%These sets induce subgraphs of the underlying graph. Let $\Gamma(t)$ be the subgraph of $G$ induced by the vacant set of the walk at step $t$. Similarly, let $\widehat \Gamma(t)$ be the subgraph of $G$ induced by the edges of the vacant net. For random $r$-regular graphs $G_r$, it was previously established that for a simple random walk, the graph $\Gamma(t)$ of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs $G_{n,p}$. Thus, for $r \ge 3$ there is an explicit value $t^*=t^*(r)$ of the walk, such that for $t\leq (1-\epsilon)t^*$, $\Gamma(t)$ has a unique giant component, plus components of size $O(\log n)$, whereas for $t\geq (1+\epsilon)t^*$ all the components of $\Gamma(t)$ are of size $O(\log n)$. We establish the threshold value $\widehat t$ for a phase transition in the graph $\widehat \Gamma(t)$ of the vacant net of a simple random walk on a random $r$-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for $r$ even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random $r$-regular graphs. The main findings are the following: As $r$ increases the threshold for the vacant set converges to $n \log r$ in all three walks. For the vacant net, the threshold converges to $rn/2 \; \log n$ for both the simple random walk and non-backtracking random walk. When $r\ge 4$ is even, the threshold for the vacant net of the unvisited edge process converges to $rn/2$, which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk

Phase transition for the vacant set left by random walk on the giant component of a random graph

Abstract. We study the simple random walk on the giant component of a supercritical Erdős-Rényi random graph on n vertices, in particular the so-called vacant set at level u, the complement of the trajectory of the random walk run up to a time proportional to u and n. We show that the component structure of the vacant set exhibits a phase transition at a critical parameter u?: For u < u? the vacant set has with high probability a unique giant component of order n and all other components small, of order at most log7 n, whereas for u> u? it has with high probability all components small. Moreover, we show that u? coincides with the critical parameter of random interlacements on a Poisson-Galton-Watson tree, which was identified in [Tas10]. 1. introduction Recently, several authors have been studying percolative properties of the vacant set left by random walk on finite graphs and the connections of this problem to the random inter-lacements model introduced in [Szn10]. The topic was initiated with the study of random walk on the d-dimensional discrete torus in [BS08], which was further investigated in [TW11]. [ČTW11], [ČT11] and [CF11] studied random walk on the random regular graph, and [CF11

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 Gn,p (above the connectivity threshold) and for random regular graphs Gr,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 Gn,p and Gr we give the value of t*, and the size of Γ(t). For Gr, 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(logn). We also show that for random digraphs Dn,p above the strong connectivity threshold, there is a similar directed phase transition. Thus fort ≤ (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).</p

Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders

We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_\star$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_\star$, and on the contrary, whenever $h<h_\star$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_\star$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [AC1].Comment: 42 pages, 1 figur