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Register a new userGlauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability
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We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).
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Consider random $d$-regular graphs, i.e., random graphs such that there are exactly $d$ edges from each vertex for some $dge 3$. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a $d$-regular graph chosen uniformly at random from the collection of all $d$-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random $d$-regular graph, both in the quenched as well as the annealed settings. Let $beta$ be the inverse temperature, $beta_c$ be the critical temperature and $B$ be the external magnetic field. Concerning the annealed measure, we show that for $beta > beta_c$ there exists $hat{B}_c(beta)in (0,infty)$ such that the model is metastable (i.e., the mixing time is exponential in the graph size $n$) when $beta> beta_c$ and $0 leq B < hat{B}_c(beta)$, whereas it exhibits the cut-off phenomenon at $c_star n log n$ with a window of order $n$ when $beta < beta_c$ or $beta > beta_c$ and $B>hat{B}_c(beta)$. Interestingly, $hat{B}_c(beta)$ coincides with the critical external field of the Ising model on the $d$-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists $B_c(beta)$ with $B_c(beta) leq hat{B}_c(beta)$ such that for $beta> beta_c$, the mixing time is at least exponential along some subsequence $(n_k)_{kgeq 1}$ when $0 leq B < B_c(beta)$, whereas it is less than or equal to $Cnlog n$ when $B>hat{B}_c(beta)$. The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.
Consider the complete graph on $n$ vertices. To each vertex assign an Ising spin that can take the values $-1$ or $+1$. Each spin $i in [n]={1,2,dots, n}$ interacts with a magnetic field $h in [0,infty)$, while each pair of spins $i,j in [n]$ interact with each other at coupling strength $n^{-1} J(i)J(j)$, where $J=(J(i))_{i in [n]}$ are i.i.d. non-negative random variables drawn from a prescribed probability distribution $mathcal{P}$. Spins flip according to a Metropolis dynamics at inverse temperature $beta in (0,infty)$. We show that there are critical thresholds $beta_c$ and $h_c(beta)$ such that, in the limit as $ntoinfty$, the system exhibits metastable behaviour if and only if $beta in (beta_c, infty)$ and $h in [0,h_c(beta))$. Our main result are sharp asymptotics, up to a multiplicative error $1+o_n(1)$, of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of $J$, while the correction terms do. The leading order of the correction term is $sqrt{n}$ times a centred Gaussian random variable with a complicated variance depending on $beta,h,mathcal{P}$ and on the metastable state. The critical thresholds $beta_c$ and $h_c(beta)$ depend on $mathcal{P}$, and so does the number of metastable states. We derive an explicit formula for $beta_c$ and identify some properties of $beta mapsto h_c(beta)$. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.
Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres conjecture in [3] relating sharp cut off, spectral gap, and mixing time.
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(alphawedge2)$ for self-avoiding walk and the Ising model, and $d>3(alphawedge2)$ for percolation, where $d$ denotes the dimension and $alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.
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