A remark on charge transfer processes in multi-particle systems

We assess the probability of resonances between sufficiently distant states in a combinatorial graph serving as the configuration space of an N-particle disordered quantum system. This includes the cases where the transition "shuffles" the particles in the configurations. In presence of a random external potential V, such pairs of configurations give rise to local (random) Hamiltonians which are strongly coupled, so that eigenvalue (or eigenfunction) correlator bounds are difficult to obtain. This difficulty, which occurs for three or more particles, results in eigenfunction decay bounds weaker than expected. We show that more efficient bounds, obtained so far only for 2-particle systems, extend to any number of particles

Poisson approximation for non-backtracking random walks

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. Alon, Benjamini, Lubetzky and Sodin studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length $n$ on a high-girth $n$-vertex regular expander, is typically $(1+o(1))\frac{\log n}{\log\log n}$, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun's sieve with some extensions of the ideas in Alon et al. Let $N_t$ denote the number of vertices visited precisely $t$ times by a non-backtracking random walk of length $n$ on a regular $n$-vertex expander of fixed degree and girth $g$. We prove that if $g=\omega(1)$, then for any fixed $t$, $N_t/n$ is typically $\frac{1}{\mathrm{e}t!}+o(1)$. Furthermore, if $g=\Omega(\log\log n)$, then $N_t/n$ is typically $\frac{1+o(1)}{\mathrm{e}t!}$ uniformly on all $t \leq (1-o(1))\frac{\log n}{\log\log n}$ and 0 for all $t \geq (1+o(1))\frac{\log n}{\log\log n}$. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.Comment: 19 page

Direct Scaling Analysis of localization in disordered systems. II. Multi-particle lattice systems

We adapt a simplified version of the Multi-Scale Analysis presented in \cite{C11} to multi-particle tight-binding Anderson models. Combined with a recent eigenvalue concentration bound for multi-particle systems \cite{C10}, the new method leads to a simple proof of the multi-particle dynamical localization with more optimal decay bounds on eigenfunctions than in \cite{CS09b,AW09a,AW09b}, for a large class of strongly mixing random potentials. All earlier results required the random potential to be IID. We also extend the result on multi-particle localization to models with a rapidly decaying interaction

Bypassing Erd\H{o}s' Girth Conjecture: Hybrid Stretch and Sourcewise Spanners

An $(\alpha,\beta)$-spanner of an $n$-vertex graph $G=(V,E)$ is a subgraph $H$ of $G$ satisfying that $dist(u, v, H) \leq \alpha \cdot dist(u, v, G)+\beta$ for every pair $(u, v)\in V \times V$, where $dist(u,v,G')$ denotes the distance between $u$ and $v$ in $G' \subseteq G$. It is known that for every integer $k \geq 1$, every graph $G$ has a polynomially constructible $(2k-1,0)$-spanner of size $O(n^{1+1/k})$. This size-stretch bound is essentially optimal by the girth conjecture. It is therefore intriguing to ask if one can "bypass" the conjecture by settling for a multiplicative stretch of $2k-1$ only for \emph{neighboring} vertex pairs, while maintaining a strictly \emph{better} multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of \emph{$k$-hybrid spanners}, in which non neighboring vertex pairs enjoy a \emph{multiplicative} $k$-stretch and the neighboring vertex pairs enjoy a \emph{multiplicative} $(2k-1)$ stretch (hence, tight by the conjecture). We show that for every unweighted $n$-vertex graph $G$ with $m$ edges, there is a (polynomially constructible) $k$-hybrid spanner with $O(k^2 \cdot n^{1+1/k})$ edges. \indent An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs $S \times V$ for a given subset of vertices $S \subseteq V$ referred to here as \emph{sources}. Spanners in which the distances in $S \times V$ are bounded are referred to as \emph{sourcewise spanners}. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more)

Cutoff for the Swendsen-Wang dynamics on the lattice

We study the Swendsen-Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen-Wang dynamics is a non-local Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work we derive strong enough bounds on the mixing time, proving that the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures exhibits a sharp transition from "unmixed" to "well-mixed," which is called the cutoff phenomenon. In particular, we establish that at high enough temperatures the Swendsen-Wang dynamics on the torus $(\mathbb{Z}/n\mathbb{Z})^d$ has cutoff at time $\frac{d}{2} \left( -\log (1-\gamma) \right)^{-1} \log n$, where $\gamma(\beta)$ is the spectral gap of the infinite-volume dynamics.Comment: 44 pages, 2 figure

Zero-one laws for binary random fields

A set of binary random variables indexed by a lattice torus is considered. Under a mixing hypothesis, the probability of any proposition belonging to the first order logic of colored graphs tends to 0 or 1, as the size of the lattice tends to infinity. For the particular case of the Ising model with bounded pair potential and surface potential tending to $-\infty$, the threshold functions of local propositions are computed, and sufficient conditions for the zero-one law are given.Comment: 16 pages, 1 figure. Keywords: zero-one law, first-order logic, random field, weak dependence, Ising mode

Myopic Models of Population Dynamics on Infinite Networks

Reaction-diffusion equations are treated on infinite networks using semigroup methods. To blend high fidelity local analysis with coarse remote modeling, initial data and solutions come from a uniformly closed algebra generated by functions which are flat at infinity. The algebra is associated with a compactification of the network which facilitates the description of spatial asymptotics. Diffusive effects disappear at infinity, greatly simplifying the remote dynamics. Accelerated diffusion models with conventional eigenfunctions expansions are constructed to provide opportunities for finite dimensional approximation.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1109.313

Homological stability for mapping class groups of surfaces

We give a complete and detailed proof of Harer's stability theorem for the homology of mapping class groups of surfaces, with the best stability range presently known. This theorem and its proof have seen several improvements since Harer's original proof in the mid-80's, and our purpose here is to assemble these many additions.Comment: Proof of claim 3 in the spectral sequence argument corrected (see new lemma 2.5, corollaries 2.6 and 2.7). To appear in the Handbook of Modul

Steiner Point Removal with Distortion O(log⁡k)O(\log k)

In the Steiner point removal (SPR) problem, we are given a weighted graph $G=(V,E)$ and a set of terminals $K\subset V$ of size $k$. The objective is to find a minor $M$ of $G$ with only the terminals as its vertex set, such that the distance between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [KKN15] used a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [Che17] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We improve the analysis of this ball-growing algorithm even further, bounding the distortion by $O(\log k)$

Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this paper, we turn the tables: our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete world.Comment: 26 pages, 6 figures. arXiv admin note: text overlap with arXiv:1501.0231