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Itai Benjamini

Publications and source records attributed to Itai Benjamini.

At least 73 records · Page 4Linked to original sources
arXiv2015

Juntas in the $\ell^{1}$-grid and Lipschitz maps between discrete tori

We show that if $A \subset [k]^n$, then $A$ is $ε$-close to a junta depending upon at most $\exp(O(|\partial A|/(k^{n-1}ε)))$ coordinates, where $\partial A$ denotes the edge-boundary of $A$ in the $\ell^1$-grid. This is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large subsets of the $\ell^1$-grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [T. Austin, On the failure of concentration for the $\ell^{\infty}$-ball, preprint]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

math.CO↗
arXiv2015

An Upper bound on the growth of Dirichlet tilings of hyperbolic spaces

It is shown that the growth rate $(\lim_r |B(r)|^{1/r})$ of any $k$ faces Dirichlet tiling of the real hyperbolic space $\mathbb{H}^d, d>2,$ is at most $k-1-ε$, for an $ε> 0$, depending only on $k$ and $d$. We don't know if there is a universal $ε_u > 0$, such that $k-1-ε_u$ upperbounds the growth rate for any $k$-regular tiling, when $ d > 2$?

math.GR↗
arXiv2015

Homogenization via sprinkling

We show that a superposition of an $\varepsilon$-Bernoulli bond percolation and any everywhere percolating subgraph of $\mathbb Z^d$, $d\ge 2$, results in a connected subgraph, which after a renormalization dominates supercritical Bernoulli percolation. This result, which confirms a conjecture of the first author together with Häggström and Schramm (2000), is mainly motivated by obtaining finite volume characterizations of uniqueness for general percolation processes.

math.PR↗
arXiv2015

First passage percolation on nilpotent Cayley graphs and beyond

Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a previous work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting a well-known argument for Z^d, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e. deterministic if and only the volume growth is subexponential.

math.PR↗
arXiv2014

Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation

We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation.

math.PR↗
arXiv2014

On the scaling limit of finite vertex transitive graphs with large diameter

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $ 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.

math.GR↗
arXiv2014

Harmonic measure in the presence of a spectral gap

We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting point is not more than a constant factor of the uniform measure on the set. For large sets there is a tight logarithmic correction factor. We also show that positive spectral gap does not allow for a fixed proportion of the harmonic measure of sets to be supported on small subsets, in contrast to the situation in Euclidean space. The results are quantitative as a function of the spectral gap, and apply also when the spectral gap decays to 0 as the size of the graph grows to infinity. As an application we consider a model of diffusion limited aggregation, or DLA, on finite graphs, obtaining upper bounds on the growth rate of the aggregate.

math.PR↗
arXiv2014

Ergodic Theory on Stationary Random Graphs

A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic function. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.

math.PR↗
arXiv2014

Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs

In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. On Erdős-Rényi random graphs with degrees $Ω(\sqrt{n})$, we show that when initial opinions are chosen i.i.d. then agents all converge to the initial majority opinion, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions.

math.PR↗
arXiv2014

Acquaintance Time of a Graph

We define the following parameter of connected graphs. For a given graph $G$ we place one agent in each vertex of $G$. Every pair of agents sharing a common edge is declared to be acquainted. In each round we choose some matching of $G$ (not necessarily a maximal matching), and for each edge in the matching the agents on this edge swap places. After the swap, again, every pair of agents sharing a common edge become acquainted, and the process continues. We define the \emph{acquaintance time} of a graph $G$, denoted by $AC(G)$, to be the minimal number of rounds required until every two agents are acquainted. We first study the acquaintance time for some natural families of graphs including the path, expanders, the binary tree, and the complete bipartite graph. We also show that for all positive integers $n$ and $k \leq n^{1.5}$ there exists an $n$-vertex graph $G$ such that $AC(G) =Θ(k)$. We also prove that for all $n$-vertex connected graphs $G$ we have $AC(G) = O\left(\frac{n^2}{\log(n)/\log\log(n)}\right)$, improving the $O(n^2)$ trivial upper bound achieved by sequentially letting each agent perform depth-first search along a spanning tree of $G$. Studying the computational complexity of this problem, we prove that for any constant $t \geq 1$ the problem of deciding that a given graph $G$ has $AC(G) \leq t$ or $AC(G) \geq 2t$ is $\mathcal{NP}$-complete. That is, $AC(G)$ is $\mathcal{NP}$-hard to approximate within multiplicative factor of 2, as well as within any additive constant factor. On the algorithmic side, we give a deterministic algorithm that given a graph $G$ with $AC(G)=1$ finds a ${\lceil n/c\rceil}$-rounds strategy for acquaintance in time $n^{c+O(1)}$. We also design a randomized polynomial time algorithm that given a graph $G$ with $AC(G)=1$ finds with high probability an $O(\log(n))$-rounds strategy for acquaintance.

cs.CC↗
arXiv2013

Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n there exists an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y) where distance(,) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive. This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who raised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions will require some new ideas beyond the sensitivity-based structural results of Boppana [IPL 97]. We study the mapping f further and show that it (and its inverse) are computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f is "approximately local" in the sense that all but the last output bit of f are essentially determined by a single input bit.

math.CO↗
arXiv2013

A generalized Polya's urn with graph based interactions

Given a finite connected graph G, place a bin at each vertex. Two bins are called a pair if they share an edge of G. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls raised by some fixed power a>0. We characterize the limiting behavior of the proportion of balls in the bins. The proof uses a dynamical approach to relate the proportion of balls to a vector field. Our main result is that the limit set of the proportion of balls is contained in the equilibria set of the vector field. We also prove that if a<1 then there is a single point v=v(G,a) with nonzero entries such that the proportion converges to v almost surely. A special case is when G is regular and a is at most 1. We show e.g. that if G is non-bipartite then the proportion of balls in the bins converges to the uniform measure almost surely.

math.PR↗
arXiv2013

Annihilation and coalescence on binary trees

An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability vector p=(p_1,...,p_{k+1}). The remaining nodes become infected or not via annihilation and coalescence: nodes whose two children have the same state (infected or not) are infected (or not) by this state; nodes whose two children have different states are not infected; nodes whose only one of the children is infected are infected by this state. In this note we characterize, for every p, the limiting distribution at the root node of the tree as the height n goes to infinity. We also consider a variant of the model when k=2 and a mutation can happen, with a fixed probability q, at each infection step. We characterize, in terms of p and q, the limiting distribution at the root node of the tree as the height n goes to infinity. The distribution at the root node is driven by a dynamical system, and the proofs rely on the analysis of this dynamics.

math.DS↗
arXiv2013

Unimodular Random Trees

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also prove that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.

math.PR↗
arXiv2013

Counting bats

We demonstrate an algorithm that reconstructs the number of walkers in an unknown graph from observations of their returns to a fixed point.

math.PR↗