We consider the discrete two-dimensional Gaussian free field on a box of side length N , with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.
We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.
We establish the satisfiability threshold for random k-sat for all k ě k0. That is, there exists a limiting density αspkq such that a random k-sat formula of clause density α is with high probability satisfiable for α ă αs, and unsatisfiable for α ą αs. The satisfiability threshold αspkq is given explicitly by the one-step replica symmetry breaking (1rsb) prediction from statistical physics. We believe that our methods may apply to a range of random constraint satisfaction problems in the 1rsb class.
We consider the random regular k-nae-sat problem with n variables each appearing in exactly d clauses. For all k exceeding an absolute constant k 0 , we establish explicitly the satisfiability threshold d ‹ " d ‹ pkq. We prove that for d ă d ‹ the problem is satisfiable with high probability while for d ą d ‹ the problem is unsatisfiable with high probability. If the threshold d ‹ lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krzakała et al. (2007). Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.
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