Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph Gn,p. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in Gn,p. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in Gn,p for all p = p(n) satisfying n −1 log n p 1. Contents 42 8. The Poisson regime 50 9. Beyond polynomials with nonnegative coefficients 58 10. Concluding remarks 59 References 61
In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with a = b = c = 1 on the square lattice), or, equivalently, of the uniform random homomorphisms from Z 2 to Z, has logarithmic variance. This establishes a strong form of roughness of this height function.
Let G be a quasi-transitive graph on V. A random field X = (Xv) v∈V whose distribution is invariant under all automorphisms of G is said to be a factor of i.i.d. if there exists an i.i.d. process Y = (Yv) v∈V and an equivariant map ϕ such that ϕ(Y ) has the same distribution as X. Such a map, also called a coding, is said to be finitary if, for every v ∈ V, there exists a finite (but random) set U ⊂ V such that ϕ(Y )v is determined by {Yu} u∈U . We construct a coding for the random-cluster model on general quasi-transitive graphs, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström-Jonasson-Lyons [18]. We also prove that the coding radius has exponential tails in the sub-critical regime. As a corollary, we obtain a finitary coding for the sub-critical Potts model on G whose coding radius has exponential tails. In the case of G = Z d , we also construct a finitary, translation-equivariant coding for the sub-critical random-cluster and Potts models using a finite-valued i.i.d. process Y . To do this, we extend a mixing-time result of to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth. Our methods also apply to any monotone model satisfying mild technical (but natural) requirements.
It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning. For the latter, we extend the recent proof by P. Lammers of the existence of a depinning transition in the integervalued Gaussian field in two-dimensional graphs of degree three to all doubly-periodic graphs, in particular to Z 2 . Taken together these two statements yield a new perspective on the Berezinskii-Kosterlitz-Thouless phase transition in the Villain model, and complete a new proof of depinning in two-dimensional integer-valued height functions.
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