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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)

- Nike Sun
- 2011

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)

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 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 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)

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