On the scaling of the chemical distance in long-range percolation models

We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{\Delta+o(1)},\qquad x,y\in C_{\infty}, |x-y|\to\infty, where \Delta^{-1} is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x-y|\to\infty. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of ``small-world'' phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.Comment: Published at http://dx.doi.org/10.1214/009117904000000577 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Scaling limit for a class of gradient fields with nonconvex potentials

We consider gradient fields $(\phi_x:x\in \mathbb{Z}^d)$ whose law takes the Gibbs--Boltzmann form $Z^{-1}\exp\{-\sum_{}V(\phi_y-\phi_x)\}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation V(\eta):=-\log\int\varrho({d}\kappa)\exp\biggl[-{1/2}\kappa\et a^2\biggr], where $\varrho$ is a positive measure with compact support in $(0,\infty)$. Hence, the potential $V$ is symmetric, but nonconvex in general. While for strictly convex $V$'s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential $V$ above scales to a Gaussian free field.Comment: Published in at http://dx.doi.org/10.1214/10-AOP548 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Eigenvalue order statistics for random Schr\"odinger operators with doubly-exponential tails

We consider random Schr\"odinger operators of the form $\Delta+\xi$, where $\Delta$ is the lattice Laplacian on $\mathbb Z^d$ and $\xi$ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of $\mathbb Z^d$. We show that for $\xi$ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where $\xi$ takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.Comment: 36 page

Forbidden gap argument for phase transitions proved by means of chessboard estimates

Chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest. In this note we show that the method not only produces a point in the phase diagram where more than one Gibbs states coexist, but that it can also be used to rule out the existence of shift-ergodic states that differ significantly from those proved to exist. For models depending on a parameter (say, the temperature), this shows that the values of the conjugate thermodynamic quantity (the energy) inside the "transitional gap" are forbidden in all shift-ergodic Gibbs states. We point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria.Comment: 26 page

Quenched invariance principle for simple random walk on percolation clusters

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.Comment: 38 pages (PTRF format) 4 figures. Version to appear in PTR

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on $\Z^4$) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability \cmss P_\omega^{2n}(0,0) after $2n$ steps is at most $C(\omega) n^{-2} \log n$, but the best lower bound till now has been $C(\omega) n^{-2}$. Here we will show that the $\log n$ term marks a real phenomenon by constructing an environment, for each sequence $\lambda_n\to\infty$, such that \cmss P_\omega^{2n}(0,0)\ge C(\omega)\log(n)n^{-2}/\lambda_n, with $C(\omega)>0$ a.s., along a deterministic subsequence of $n$'s. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the $d\ge5$ cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.Comment: 28 pages, version to appear in J. Lond. Math. So