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Gady Kozma

Publications and source records attributed to Gady Kozma.

At least 91 records · Page 5Linked to original sources
arXiv2011

Singular distributions, dimension of support, and symmetry of Fourier transform

We study the "Fourier symmetry" of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (1) A one-side extension of Frostman's theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of the support; (2) A construction of compacts of "critical" size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l^2. We also give examples of non-symmetry which may occur for measures with "small" support. A number of open questions are stated.

math.CA↗
arXiv2010

Nonamenable Liouville Graphs

Add 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.

math.MG↗
arXiv2010

A balanced excited random walk

The 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 that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed.

math.PR↗
arXiv2009

A note about critical percolation on finite graphs

In 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 that its diameter is n^{1/3} and that the simple random walk takes n steps to mix on it. Our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus Z_n^d (with d large and n tending to infinity) and the Hamming cube {0,1}^n.

math.PR↗
arXiv2009

The Alexander-Orbach conjecture holds in high dimensions

We 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 walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.

math.PR↗
arXiv2009

Arm exponents in high dimensional percolation

We 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 this probability decays like 1/r^2. Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r^2k.

math.PR↗
arXiv2009

One-dimensional long-range diffusion-limited aggregation I

We 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 conditions on the tail of the step distribution, we prove that the diameter grows as n^{beta+o(1)}, with an explicitly given beta. The growth rate of the aggregate is shown to have three phase transitions, when the walk steps have finite third moment, finite variance, and, conjecturally, finite half moment.

math.PR↗
arXiv2009

Entropy of Random Walk Range

We 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 of the trace.

math.PR↗
arXiv2008

On the precision of the spectral profile

We examine the spectral profile bound of Goel, Montenegro and Tetali for the uniform mixing time of continuous-time random walk in reversible settings. We find that it is precise up to a log log factor, and that this log log factor cannot be improved.

math.PR↗
arXiv2007

Many Random Walks Are Faster Than One

We 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 expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

math.PR↗
arXiv2007

Anomalous heat-kernel decay for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $ω_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $ω_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2n$-step return probability $P_ω^{2n}(0,0)$. We prove that $P_ω^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.

math.PR↗
arXiv2006

On the hyperplane conjecture for random convex sets

Let 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 conjecture for the class of gaussian random polytopes.

math.MG↗
arXiv2006

One cannot hear the winding number

We 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.

math.CA↗
arXiv2006

Excited random walk against a wall

We 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.

math.PR↗
arXiv2006

Percolation, Perimetry, Planarity

Let 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.

math.PR↗
arXiv2006

On the connectivity of the Poisson process on fractals

For 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 probability (log n/n)^(1/d).

math.PR↗