Geometric Influences II: Correlation Inequalities and Noise Sensitivity

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.Comment: 20 page

Stochastic domination: the contact process, Ising models and FKG measures

We prove for the contact process on $Z^d$, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate $\lambda$ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for ``downward'' FKG measures. As a consequence of this general result, we show that the plus and minus states for the Ising model on $Z^d$ dominate the same set of product measures. We show that this latter fact fails completely on the homogenous 3-ary tree. We also provide a different distinction between $Z^d$ and the homogenous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on $Z^d$ are never stochastically ordered, on the homogenous 3-ary tree, almost the complete opposite is the case. Next, we show that on $Z^d$, the set of product measures which the plus state for the Ising model dominates is strictly increasing in the temperature. Finally, we obtain a necessary and sufficient condition for a finite number of variables, which are both FKG and exchangeable, to dominate a given product measure.Comment: 27 page

A quantitative Burton-Keane estimate under strong FKG condition

We consider translationally-invariant percolation models on $\mathbb{Z}^d$ satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance $n$ (this corresponds to a finite size version of the celebrated Burton-Keane [Comm. Math. Phys. 121 (1989) 501-505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincar\'{e} inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight $q\ge1$.Comment: Published at http://dx.doi.org/10.1214/15-AOP1049 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star--triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the $q>1$ random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.Comment: 24 pages, 1 figur

Brownian motion and Random Walk above Quenched Random Wall

We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_n\}$ and $\{W_n\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N} B_n \geq W_n|W) = N^{-\gamma + o(1)}$ for a non-random $\gamma\geq 1/2$. In the classical setting, $W_n \equiv 0$, it is well-known that $\gamma = 1/2$. We prove that for any non-trivial $W$ one has $\gamma>1/2$ and the exponent $\gamma$ depends only on $\text{Var}(B_1)/\text{Var}(W_1)$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.Comment: To appear in Ann. Inst. Henri Poincar\'e Probab. Sta

On log concavity for order-preserving and order-non-reversing maps of partial orders

Stanley used the Aleksandrov-Fenchel inequalities from the theory of nixed volumes to prove the following result. Let P be a partially ordered set with n elements, and let x ∊ P. If Ni* is the number of linear extensions , ⋋ : P + (1 , 2,...,n) satisfying ⋋ (x) = i, then the sequence N*1,…,N*n is log concave (and therefore unimodal). Here the analogous results for both order-preserving and order-non-reversing maps are proved using an explicit injection. Further, if vc is the number of order-preserving maps of P into a chain of length c, then vc is shown to be 1-og concave, and the corresponding result is established for order-non-reversing maps