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Register a new userOn the extension of the FKG inequality to $n$ functions
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The 1971 Fortuin-Kasteleyn-Ginibre (FKG) inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008 one of us (Sahi) conjectured an extended version of this inequality for all $n>2$ monotone functions on a distributive lattice. Here we prove the conjecture for two special cases: for monotone functions on the unit square in ${mathbb R}^k$ whose upper level sets are $k$-dimensional rectangles, and, more significantly, for arbitrary monotone functions on the unit square in ${mathbb R}^2$. The general case for ${mathbb R}^k, k>2$ remains open.
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Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $dgeq3$. We prove that, with probability $1-N^{-1+{varepsilon}}$ for any ${varepsilon} >0$, the following two properties hold as $N to infty$ provided that $dgeq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $lambda_2, |lambda_N|leq 2+N^{-c}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized.
We introduce a formula for translating any upper bound on the percolation threshold of a lattice g into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best known asymptotic bounds on $a(mathbb{Z}^d)$ as $dto infty$. Our formula remains valid if instead of lattice animals we enumerate certain sub-species called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $pin (0,1)$.
Our work deals with symmetric rational functions and probabilistic models based on the fully inhomogeneous six vertex (ice type) model satisfying the free fermion condition. Two families of symmetric rational functions $F_lambda,G_lambda$ are defined as certain partition functions of the six vertex model, with variables corresponding to row rapidities, and the labeling signatures $lambda=(lambda_1ge ldotsge lambda_N)in mathbb{Z}^N$ encoding boundary conditions. These symmetric functions generalize Schur symmetric polynomials, as well as some of their variations, such as factorial and supersymmetric Schur polynomials. Cauchy type summation identities for $F_lambda,G_lambda$ and their skew counterparts follow from the Yang-Baxter equation. Using algebraic Bethe Ansatz, we obtain a double alternant type formula for $F_lambda$ and a Sergeev-Pragacz type formula for $G_lambda$. In the spirit of the theory of Schur processes, we define probability measures on sequences of signatures with probability weights proportional to products of our symmetric functions. We show that these measures can be viewed as determinantal point processes, and we express their correlation kernels in a double contour integral form. We present two proofs: The first is a direct computation of Eynard-Mehta type, and the second uses non-standard, inhomogeneo
We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a(mathbb{Z}^d)$ as $dto infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(mathbb{Z}^d)$. We obtain the rigorous lower bound $dot{p}_c(mathbb{Z}^3)>0.2522$ for 3-dimensional site percolation.
We introduce the class of {em strongly Rayleigh} probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons recent results on determinantal measures and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.
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