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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.
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
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
We consider a model of language development, known as the naming game, in which agents invent, share and then select descriptive words for a single object, in such a way as to promote local consensus. When formulated on a finite and connected graph,
Red and blue particles are placed in equal proportion through-out either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We compare the time it
We obtain scaling limit results for asymmetric trap models and their infinite volume counterparts, namely asymmetric K processes. Aging results for the latter processes are derived therefrom.