Abstract:
In this paper, we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connecting radius tending to infinity. We obtain that for any infection rate $\lambda \textgreater{}0$, the contact process on these graphs survives a time super-exponential in the number of vertices.

Abstract:
Let $\mathcal{C}_1$ be the largest component of the Erd\H{o}s--R\'{e}nyi random graph $\mathcal{G}(n,p)$. The mixing time of random walk on $\mathcal {C}_1$ in the strictly supercritical regime, $p=c/n$ with fixed $c>1$, was shown to have order $\log^2n$ by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, $p=(1+\varepsilon)/n$ where $\lambda=\varepsilon^3n$ is bounded, Nachmias and Peres proved that the mixing time on $\mathcal{C}_1$ is of order $n$. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of $\mathcal{C}_1$ in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim. In this paper, we show that for $p=(1+\varepsilon)/n$ with $\lambda=\varepsilon^3n\to\infty$ and $\lambda=o(n)$, the mixing time on $\mathcal{C}_1$ is with high probability of order $(n/\lambda)\log^2\lambda$. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., $p=(1-\varepsilon)/n$ with $\lambda$ as above].

Abstract:
We show that the total variation mixing time of the simple random walk on the giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are "decorated expanders" - an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations.

Abstract:
Suppose that $\CG$ is a finite, connected graph and $X$ is a lazy random walk on $\CG$. The lamplighter chain $X^\diamond$ associated with $X$ is the random walk on the wreath product $\CG^\diamond = \Z_2 \wr \CG$, the graph whose vertices consist of pairs $(f,x)$ where $f$ is a labeling of the vertices of $\CG$ by elements of $\Z_2$ and $x$ is a vertex in $\CG$. There is an edge between $(f,x)$ and $(g,y)$ in $\CG^\diamond$ if and only if $x$ is adjacent to $y$ in $\CG$ and $f(z) = g(z)$ for all $z \neq x,y$. In each step, $X^\diamond$ moves from a configuration $(f,x)$ by updating $x$ to $y$ using the transition rule of $X$ and then sampling both $f(x)$ and $f(y)$ according to the uniform distribution on $\Z_2$; $f(z)$ for $z \neq x,y$ remains unchanged. We give matching upper and lower bounds on the uniform mixing time of $X^\diamond$ provided $\CG$ satisfies mild hypotheses. In particular, when $\CG$ is the hypercube $\Z_2^d$, we show that the uniform mixing time of $X^\diamond$ is $\Theta(d 2^d)$. More generally, we show that when $\CG$ is a torus $\Z_n^d$ for $d \geq 3$, the uniform mixing time of $X^\diamond$ is $\Theta(d n^d)$ uniformly in $n$ and $d$. A critical ingredient for our proof is a concentration estimate for the local time of random walk in a subset of vertices.

Abstract:
Let $\mathcal{C}_1$ denote the largest connected component of the critical Erd\H{o}s--R\'{e}nyi 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 $p(d-1)\le1+O(n^{-1/3})$.

Abstract:
Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H\wr G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z for z different of x,y remains unchanged. We estimate the mixing time of X* in terms of the parameters of H and G. Further, we show that the relaxation time of X* is the same order as the maximal expected hitting time of G plus |G| times the relaxation time of the chain on H.

Abstract:
We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if \[(d-1)\tanh\beta<1,\] then there exists a constant C such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of n vertices and maximal degree d, where all interactions are bounded by $\beta$, and arbitrary external fields are bounded by $Cn\log n$. Moreover, the spectral gap is uniformly bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d. We further show that when $d\tanh\beta<1$, with high probability over the Erdos-Renyi random graph $G(n,d/n)$, it holds that the mixing time of Gibbs samplers is \[n^{1+\Theta({1}/{\log\log n})}.\] Both results are tight, as it is known that the mixing time for random regular and Erdos-Renyi random graphs is, with high probability, exponential in n when $(d-1)\tanh\beta>1$, and $d\tanh\beta>1$, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.

Abstract:
We study the mixing time of random graphs in the $d$-dimensional toric unit cube $[0,1]^d$ generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights, drawn from some distribution. The connectivity threshold for GTGs is comparable to that of RGGs, essentially corresponding to a connectivity radius of $r=(\log n/n)^{1/d}$. However, the degree distributions at this threshold are quite different: in an RGG the degrees are essentially uniform, while RGGs have heterogeneous degrees that depend upon the weight distribution. Herein, we study the mixing times of random walks on $d$-dimensional GTGs near the connectivity threshold for $d \geq 2$. If the weight distribution function decays with $\mathbb{P}[W \geq x] = O(1/x^{d+\nu})$ for an arbitrarily small constant $\nu>0$ then the mixing time of GTG is $\mixbound$. This matches the known mixing bounds for the $d$-dimensional RGG.

Abstract:
We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growing with the inverse temperature. In addition, we construct a graph in this class, for which the mixing time in the free boundary case is exponentially large in n. This provides a first example where the mixing time jumps from exponential to linear in n while passing from free to (+)-boundary condition. These results extend the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable graphs.

Abstract:
The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but in many situations partial information of the graph is already known beforehand. A natural question to ask is what would be a typical random graph drawn from an exponential model subject to certain constraints? In particular, will there be a similar phase transition phenomenon as that which occurs in the unconstrained exponential model? We present some general results for the constrained model and then apply them to get concrete answers in the edge-triangle model.