Results for "Gady Kozma"
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A note about critical percolation on finite graphsSep 24 2009Nov 16 2009In this note we study the geometry of the largest component C_1 of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. There it is shown that this component is of size n^{2/3}, and here we show ... More One cannot hear the winding numberDec 07 2006We construct an example of two continuous maps f and g of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis. Counting batsApr 10 2013We demonstrate an algorithm that reconstructs the number of walkers in an unknown graph from observations of their returns to a fixed point. A Resistance Bound via an Isoperimetric InequalityDec 23 2002Jun 09 2012An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented. Uniqueness of percolation on products with ZMay 13 2011Jul 28 2012We show that there exists a connected graph G with subexponential volume growth such that critical percolation on the product of G with the line has infinitely many infinite clusters. We also give some conditions under which this cannot occur. An "Analytic" Version of Menshov's Representation TheoremDec 01 2005Every measurable function f on the circle can be represented as a sum of harmonics with positive spectrum, converging in measure. For convergence almost everywhere this is not true. We discuss several other subsets of Z for which one might get a Menshov ... More Perturbing PLAFeb 24 2012We proved earlier that every measurable function on the circle, after a uniformly small perturbation, can be written as a power series (i.e. a series of exponentials with positive frequencies), which converges almost everywhere. Here we show that this ... More Upper bounds on the percolation correlation lengthFeb 08 2019We study the size of the near-critical window for Bernoulli percolation on $\mathbb Z^d$. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by ... More Excited random walk against a wallSep 21 2005Oct 27 2006We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of square-root log n. Localization for Linearly Edge Reinforced Random WalksMar 19 2012We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial ... More The mixing time of the giant component of a random graphOct 15 2006Jul 31 2016We show that the total variation mixing time of the simple random walk on the giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure ... More Entropy of Random Walk RangeMar 18 2009We study the entropy of the set traced by an $n$-step random walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are essentially governed by the size of the boundary ... More Discrete curvature and abelian groupsJan 03 2015Oct 23 2015We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. An appealing feature of this discrete version seems to be that it is fairly straightforward to compute this notion of curvature ... More Percolation of finite clusters and infinite surfacesMar 07 2013Oct 16 2013Two related issues are explored for bond percolation on the d-dimensional cubic lattice (with d > 2) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? ... More Disorder, entropy and harmonic functionsNov 21 2011Oct 28 2015We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb{Z}^d$. We prove that the vector space of harmonic functions growing at most linearly is $(d+1)$-dimensional ... More Percolation, Perimetry, PlanaritySep 10 2005Apr 11 2006Let G be a planar graph with polynomial growth and isoperimetric dimension bigger than 1. Then the critical p for Bernoulli percolation on G satisfies p<1. On removing one point from a compact spaceApr 15 2005If B is a compact space and B\{pt} is Lindelof then B^k\{pt} is star-Linedlof for every cardinality k. If B\{pt} is compact then B^k\{pt} is discretely star-Lindelof. In particular, this gives new examples of Tychonoff discretely star-Lindelof spaces ... More Singular distributions and symmetry of the spectrumJan 12 2013This is a survey of the "Fourier symmetry" of measures and distributions on the circle in relation with the size of their support. Mostly it is based on our paper arxiv:1004.3631 and a talk given by the second author in the 2012 Abel symposium. Combining Riesz basesOct 23 2012Apr 15 2014We show that any finite union of intervals supports a Riesz basis of exponentials Random Menshov spectraOct 20 2005We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely. Menshov representation spectraOct 28 2005A Menshov spectrum is a subset of the integers that is sufficient for representing every measurable function as an almost-everywhere converging trigonometric (non-Fourier) sum. In this language the celebrated "Menshov representation theorem" states that ... More Continuous vs discrete spins in the hyperbolic planeSep 28 2016Oct 17 2016We study the $O(n)$ model on planar hyperbolic cocompact lattices, with free boundary conditions. We observe that the pair correlations decay exponentially with distance, for all temperatures, if and only if $n>1$. A null series with small anti-analytic partOct 19 2005We show that it is possible for a square integrable function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space. A consequence in the uniqueness theory is ... More Is PLA large?Oct 06 2005Oct 18 2005We examine the class of functions representable by an analytic sum (by which we mean a trigonometric sum involving only positive frequencies) converging almost everywhere. We show that it is dense but that it is first category and has zero Wiener measure. ... More Ordering the representations of S_n using the interchange processMar 08 2010Jul 21 2011Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett and Richthammer, we define an associated order on the irreducible representations of S_n. Aldous' conjecture is equivalent to certain ... More On the gaps between zeros of trigonometric polynomialsJan 02 2006We show that for every finite symetric set S of integer vectors, every real trigonometric polynomial on the d dimensional torus with spectrum in S has a zero in every closed ball of diameter D, where D is the sum over S of 1 over 4 times the L2 norm of ... More Nonamenable Liouville GraphsOct 16 2010Add to each level of binary tree edges to make the induced graph on the level a uniform expander. It is shown that such a graph admits no non-constant bounded harmonic functions. Divisibility and Laws in Finite Simple GroupsMar 10 2014May 20 2014We provide new bounds for the divisibility function of the free group F_2 and construct short laws for the symmetric groups Sym(n). The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length ... More Groups with minimal harmonic functions as small as you likeMay 24 2016For any order of growth $f(n)=o(\log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function ... More Arm exponents in high dimensional percolationNov 04 2009We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We prove that ... More The Alexander-Orbach conjecture holds in high dimensionsJun 09 2008Nov 16 2009We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random ... More The probability of long cycles in interchange processesSep 20 2010May 25 2012We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange ... More I knew I should have taken that left turn at AlbuquerqueAug 25 2010May 02 2011We study the Laplacian-infinity path as an extreme case of the Laplacian-alpha random walk. Although, in the finite alpha case, there is reason to believe that the process converges to SLE, we show that this is not the case when alpha is infinite. In ... More Irreducible polynomials of bounded heightOct 14 2017The goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\ldots, 210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove that the Galois ... More On the hyperplane conjecture for random convex setsDec 18 2006Let N > n, and denote by K the convex hull of N independent standard gaussian random vectors in an n-dimensional Euclidean space. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane ... More The Toom Interface Via CouplingJan 20 2015Sep 19 2016We consider a one dimensional interacting particle system which describes the effective interface dynamics of the two dimensional Toom model at low temperature and noise. We prove a number of basic properties of this model. First we consider the dynamics ... More Random walks with $k$-wise independent incrementsMay 31 2004We construct examples of a random walk with pairwise-independent steps which is almost-surely bounded, and for any $m$ and $k$ a random walk with $k$-wise independent steps which has no stationary distribution modulo $m$. A balanced excited random walkSep 03 2010The following random process on $\Z^4$ is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove ... More Supercritical self-avoiding walks are space-fillingOct 13 2011Sep 25 2012We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We show that the random trajectory becomes space-filling ... More On the connectivity of the Poisson process on fractalsApr 11 2006For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu is, with high ... More The Toom Interface Via CouplingJan 20 2015Jan 13 2018We consider a one dimensional interacting particle system which describes the effective interface dynamics of the two dimensional Toom model at low temperature and noise. We prove a number of basic properties of this model. First we consider the dynamics ... More The minimal spanning tree and the upper box dimensionNov 26 2003Nov 30 2003We show that the alpha-weight of an MST over n points in a metric space with upper box dimension d has a bound independent of n if alpha is smaller than d and does not have one if alpha is larger than d. Waiting for a bat to fly by (in polynomial time)Oct 28 2003We observe returns of a simple random walk on a finite graph to a fixed node, and would like to infer properties of the graph, in particular properties of the spectrum of the transition matrix. This is not possible in general, but at least the eigenvalues ... More One-dimensional long-range diffusion-limited aggregation IOct 23 2009We examine diffusion-limited aggregation generated by a random walk on Z with long jumps. We derive upper and lower bounds on the growth rate of the aggregate as a function of the number moments a single step of the walk has. Under various regularity ... More Brochette percolationAug 17 2016Apr 20 2017We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $\mathbb{Z}$. ... More Lace expansion for dummiesDec 04 2015Apr 26 2016We show Green's function asymptotic upper bound for the two-point function of weakly self-avoiding walk in dimension bigger than 4, revisiting a classic problem. Our proof relies on Banach algebras to analyse the lace-expansion fixed point equation and ... More Random Walk in Changing EnvironmentApr 19 2015May 25 2015In this paper we introduce the notion of \emph{Random Walk in Changing Environment} - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and edge ... More Brochette percolationAug 17 2016We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $\mathbb{Z}$. ... More Uniqueness of the infinite noodleJan 04 2017Jan 23 2017Consider the graph obtained by superposition of an independent pair of uniform infinite non-crossing perfect matchings of the set of integers. We prove that this graph contains at most one infinite path. Several motivations are discussed. Minimal growth harmonic functions on lamplighter groupsJul 04 2016We study the minimal possible growth of harmonic functions on lamplighters. We find that $(\mathbb{Z}/2)\wr \mathbb{Z}$ has no sublinear harmonic functions, $(\mathbb{Z}/2)\wr \mathbb{Z}^2$ has no sublogarithmic harmonic functions, and neither has the ... More Seven-dimensional forest firesFeb 27 2013Jul 03 2015We show that in high dimensional Bernoulli percolation, removing from a thin infinite cluster a much thinner infinite cluster leaves an infinite component. This observation has implications for the van den Berg-Brouwer forest fire process, also known ... More Minimum Average Distance TriangulationsDec 08 2011Jun 20 2012We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x and y along ... More Modeling supernova emission at late timesMar 26 1999We compare model calculations with observations of supernovae at late times to infer the time evolution of temperature, ionization and line emission. Here we mainly report on our results from our modeling of SN 1987A. We discuss the oxygen mass from the ... More Diffusion of a Deformable Body in a random FlowApr 19 2001Oct 03 2001We consider a deformable body immersed in an incompressible liquid that is randomly stirred. Sticking to physical situations in which the body departs only slightly from its spherical shape, we calculate the diffusion constant of the body. We give explicitly ... More Diffusion of a Deformable Body in a Randomly Stirred Host FluidOct 03 2001Consider a deformable body immersed in an incompressible liquid that is randomly stirred. Sticking to physical situations in which the body departs only slightly from its spherical shape, we investigate the motion of the body, calculate its mean squared ... More Many Random Walks Are Faster Than OneMay 03 2007Nov 20 2007We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the ... More Maximum Scatter TSP in Doubling MetricsDec 09 2015Jun 28 2016We study the problem of finding a tour of $n$ points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as Maximum Scatter ... More Streaming Algorithms for Partitioning Integer SequencesApr 07 2014Jul 07 2014We study the problem of partitioning integer sequences in the one-pass data streaming model. Given is an input stream of integers $X \in \{0, 1, \dots, m \}^n$ of length $n$ with maximum element $m$, and a parameter $p$. The goal is to output the positions ... More The Phase Transition for Dyadic TilingsJul 13 2011Jul 20 2012A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently ... More Late Spectral Evolution of SN 1987A: II. Line EmissionDec 17 1997Using the temperature and ionization calculated in our previous paper, we model the spectral evolution of SN 1987A. The IR-catastrophe is seen in the metal lines as a transition from thermal to non-thermal excitation, most clearly in the [O I] 6300, 6364 ... More Consensus formation on adaptive networksJul 30 2007The structure of a network can significantly influence the properties of the dynamical processes which take place on them. While many studies have been devoted to this influence, much less attention has been devoted to the interplay and feedback mechanisms ... More Stochastic Growth in a Small WorldMay 01 2003We considered the Edwards-Wilkinson model on a small-world network. We studied the finite-size behavior of the surface width by performing exact numerical diagonalization for the underlying coupling matrix. We found that the spectrum exhibits a gap or ... More Shattering, Graph Orientations, and ConnectivityNov 06 2012We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph ... More Radioactivities and nucleosynthesis in SN 1987ADec 17 2001The nucleosynthesis and production of radioactive elements in SN 1987A are reviewed. Different methods for estimating the masses of 56Ni, 57Ni, and 44Ti are discussed, and we conclude that broad band photometry in combination with time-dependent models ... More Finsler connection for general Lagrangian systemsAug 11 2014Feb 24 2016We give a Finsler non-linear connection by a new simplified definition for not only regular case but also singular case. In regular case, it corresponds to non-linear connection part of Berwald's connection, but our connection is expressed not in line ... More New Horoball Packing Density Lower Bound in Hyperbolic 5-spaceSep 13 2018We describe the optimal horoball packings of asymptotic Koszul type Coxeter simplex tilings of $5$-dimensional hyperbolic space where the symmetries of the packings are generated by Coxeter groups. We find that the optimal horoball packing density of ... More The High Eccentricity of the Planet Around 16 Cyg BNov 18 1996We consider the high eccentricity, 0.66, of the newly discovered planet around 16 Cyg B, using the fact that the parent star is part of a wide binary. We show that the high eccentricity of the planet could be the result of tidal forces exerted on 16 Cyg ... More Why did Supernova 1054 shine at late times?Dec 15 2000The Crab nebula is the remnant of supernova 1054 (SN 1054). The progenitor of this supernova has, based on nucleosynthesis arguments, been modeled as an 8-10 solar mass star. Here we point out that the observations of the late light curve of SN 1054, ... More Diffusion Processes on Power-Law Small-World NetworksJan 20 2005May 20 2005We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents. The ... More The 44Ti-powered spectrum of SN 1987AMar 18 2011May 25 2011SN 1987A provides a unique opportunity to study the evolution of a supernova from explosion into very late phases. Due to the rich chemical structure, the multitude of physical process involved, and extensive radiative transfer effects, detailed modeling ... More