A note on Talagrand's variance bound in terms of influences

Let X_1,..., X_n be independent Bernoulli random variables and $f$ a function on {0,1}^n. In the well-known paper (Talagrand1994) Talagrand gave an upper bound for the variance of f in terms of the individual influences of the X_i's. This bound turned out to be very useful, for instance in percolation theory and related fields. In many situations a similar bound was needed for random variables taking more than two values. Generalizations of this type have indeed been obtained in the literature (see e.g. (Cordero-Erausquin2011), but the proofs are quite different from that in (Talagrand1994). This might raise the impression that Talagrand's original method is not sufficiently robust to obtain such generalizations. However, our paper gives an almost self-contained proof of the above mentioned generalization, by modifying step-by-step Talagrand's original proof.Comment: 10 page

Sublinearity of the travel-time variance for dependent first-passage percolation

Let $E$ be the set of edges of the $d$-dimensional cubic lattice $\mathbb{Z}^d$, with $d\geq2$, and let $t(e),e\in E$, be nonnegative values. The passage time from a vertex $v$ to a vertex $w$ is defined as $\inf_{\pi:v\rightarrow w}\sum_{e\in\pi}t(e)$, where the infimum is over all paths $\pi$ from $v$ to $w$, and the sum is over all edges $e$ of $\pi$. Benjamini, Kalai and Schramm [2] proved that if the $t(e)$'s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex $v$ is sublinear in the distance from 0 to $v$. This result was extended to a large class of independent, continuously distributed $t$-variables by Bena\"{\i}m and Rossignol [1]. We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the $t(e)$'s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical "Ising landscape."Comment: Published in at http://dx.doi.org/10.1214/10-AOP631 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A percolation process on the binary tree where large finite clusters are frozen

We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the process converges in some sense to the frozen percolation process introduced by Aldous. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin.Comment: 11 page

The regional structure of retail sector in the Northern Hungary Region

The retail sector plays a decisive role in the development of urban-rural relations. As it is not only a narrower segment of the population that participates in it and it is a good measure of the central role of cities on a market basis. Our study is part of a broader research project covering the North Hungarian region, where we investigate market access opportunities for local producers through short supply chains. As a basis for the primary research of a greater project we conducted secondary research based on the databases of the Hungarian Central Statistical Office. In addition to the number of retail stores, we investigated the possible impacts of changes in population and incomes on demand for retail stores and centres in each city and their surrounding rural areas. In terms of the socio-economic sense, the region shows a less favourable picture than the national average, which also manifests itself in the lower number of retail units. However, in the region, significant differences can also be observed in the settlement hierarchy and spatial characteristics. There is some duality in the fact that the population per unit of trade is relatively high, but at the same time, the number of units in the examined chain stores is relatively smaller in the region

Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters

Frozen percolation on the binary tree was introduced by Aldous [1], inspired by sol-gel transitions. We investigate a version of the model on the triangular lattice, where connected components stop growing (freeze) as soon as they contain at least N vertices, where N is a (typically large) parameter. For the process in certain +nite domains, we show a Òseparation of scalesÓ and use this to prove a deconcentration property. Then, for the full-plane process, we prove an accurate comparison to the process in suitable +nite domains, and obtain that, with high probability (as N→), the origin belongs in the nal con+guration to a mesoscopic cluster, i.e., a cluster which contains many, but much fewer than N, vertices (and hence is non-frozen). For this work we develop new interesting properties for near-critical percolation, including asymp-totic formulas involving the percolation probability θ(p) and the characteristic length L(p) as p → pc