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The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is np-hard to approximate the partition function or approximately sample from the… (More)

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree T , and study the existence of the free energy density φ, the limit of the log-partition function divided by the number of vertices n as n tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to… (More)

We provide an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a… (More)

This is an introductory account of the emergence of confor-mal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLEκ), a… (More)

- Amir Dembo, Nike Sun
- 2012

J o u r n a l o f P r o b a b i l i t y Electron. Abstract Let T be a rooted supercritical multi-type Galton–Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The λ-biased random walk (Xt) t≥0 on T is the nearest-neighbor random walk which, when at a vertex v with dv offspring, moves closer to the root with… (More)

We establish the satisfiability threshold for random k-SAT for all k ≥ k<sub>0</sub>. That is, there exists a limiting density α<sub>s</sub>(k) such that a random k-SAT formula of clause density α is with high probability satisfiable for α < α<sub>s</sub>, and unsatisfiable for α > α<sub>s</sub>. The satisfiability… (More)

We consider the random regular <i>k</i>-nae-sat problem with <i>n</i> variables each appearing in exactly <i>d</i> clauses. For all <i>k</i> exceeding an absolute constant <i>k</i><sub>0</sub>, we establish explicitly the satisfiability threshold <i>d</i><sub>*</sub> ∈ <i>d</i><sub>*</sub>(<i>k</i>). We prove that for <i>d</i> <… (More)

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the… (More)

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