We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $α$-power of jump length. The case $α=1$ corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for $α\in(0,d)$, that the random walk confined to a cubic box of side $L$ has a.s. Cheeger constant of order at least $L^{-1}$ and mixing time of order $L^2$. For the Poisson point process, we prove that at $α=d$, there is a transition from diffusive to subdiffusive behavior of the mixing time.
In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities.
We extend generalized isoperimetric-type inequalities to iterated Brownian motion over several domains in $\RR{R}^{n}$. These kinds of inequalities imply in particular that for domains of finite volume, the exit distribution and moments of the first exit time for iterated Brownian motion are maximized with the ball $D^{*}$ centered at the origin, which has the same volume as $D$
In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, the polar for $L_p$ centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Böröczky.
In this paper we adapt the well-estabilished $Γ$-calculus techniques to the context of $RCD(K,\infty)$ spaces, proving Bobkov's local isoperimetric inequality and, when $K$ is positive, the Gaussian isoperimetric inequality in this class of spaces. The proof relies on the measure-valued $Γ_2$ operator introduced by Savaré.
Let $(E,\F,μ)$ be a $\si$-finite measure space. For a non-negative symmetric measure $J(\d x, \d y):=J(x,y) \,μ(\d x)\,μ(\d y)$ on $E\times E,$ consider the quadratic form $$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 \, J(\d x,\d y)$$ in $L^2(μ)$. We characterize the relationship between the isoperimetric inequality and the super Poincaré inequality associated with $\E$. In particular, sharp Orlicz-Sobolev type and Poincaré type isoperimetric inequalities are derived for stable-like Dirichlet forms on $\R^n$, which include the existing fractional isoperimetric inequality as a special …
This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capaci…
The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetri…
In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we extend their result to all symmetric log-concave measures μon the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter), the intervals or complements of intervals have minimal perimeter.
We prove upper and lower bounds of the heat kernel for the operator $Δ-\nabla (\frac{1}{|x|^α})\cdot \nabla $ in $\mathbb{R}^{n}\setminus\{0} $ where $α>0$. We obtain these bounds from an isoperimetric inequality for a measure $\mathrm{e}^{-\frac{1}{|x|^α}}dx$ on $\mathbb{R}^{n}\setminus \{0\}$. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger's inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and …
We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em pocket-extension} of a group $G$---this leads to the study of certain finitely generated subgroups of the full permutation group $\mathbb S(G\cup \{*\})$. Some sharp estimates are obtained while many challenging questions remain.
We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.
We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least $7$ have site percolation threshold bounded away from $1/2$, which was conjectured by Benjamini and Schramm, and make progress on a…
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in \cite{Bau2}. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is …
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided $L^\infty$ bound on the ratio between their densities, Wasserstein distances, and Kullback--Leibler divergence. In particular, an extension of the Hol…
The concentration of measure prenomenon roughly states that, if a set $A$ in a product $Ω^N$ of probability spaces has measure at least one half, ``most'' of the points of $Ω^N$ are ``close'' to $A$. We proceed to a systematic exploration of this phenomenon. The meaning of the word ``most'' is made rigorous by isoperimetric-type inequalities that bound the measure of the exceptional sets. The meaning of the work ``close'' is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, …
Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomially in the fraction of the volume taken up by the kernel of the star-shaped body. The analysis is based on a new isoperimetric inequality. Our main technical contribution is a tool for proving such inequalities when the domain is not convex. As a consequence, we obtain a polynomial algorithm for computing the …
Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $|\cdot |$. We study the behavior of the probability of return for random walks driven by symmetric measures $μ$ that are such that $\sum ρ(|x|)μ(x)<\infty$ for increasing regularly varying or slowly varying functions $ρ$, for instance, $s\mapsto (1+s)^α$, $α\in (0,2]$, or $s\mapsto (1+\log (1+s))^ε$, $ε>0$. For this purpose we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to …
