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Register a new userRandom graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time
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For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.
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Given a random word of size $n$ whose letters are drawn independently from an ordered alphabet of size $m$, the fluctuations of the shape of the random RSK Young tableaux are investigated, when $n$ and $m$ converge together to infinity. If $m$ does not grow too fast and if the draws are uniform, then the limiting shape is the same as the limiting spectrum of the GUE. In the non-uniform case, a control of both highest probabilities will ensure the convergence of the first row of the tableau toward the Tracy--Widom distribution.
Finding a ground state of a given Hamiltonian on a graph $G=(V,E)$ is an important but hard problem. One of the potential methods is to use a Markov chain Monte Carlo to sample the Gibbs distribution whose highest peaks correspond to the ground states. In this short paper, we investigate the stochastic cellular automata, in which all spins are updated independently and simultaneously. We prove that (i) if the temperature is sufficiently high and fixed, then the mixing time is at most of order $log|V|$, and that (ii) if the temperature drops in time $n$ as $1/log n$, then the limiting measure is uniformly distributed over the ground states.
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