Gady Kozma

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 probablistic 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.

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

We consider the nearest-neighbor simple random walk on

Arm exponents in high dimensional percolation

\Z^d

Disorder, entropy and harmonic functions

,

Localization for linearly edge reinforced random walks

d\ge2

Combining Riesz bases

, driven by a field of bounded random conductances

Discrete curvature and abelian groups

\omega_{xy}\in[0,1]

A null series with small anti-analytic part

. The conductance law is i.i.d. subject to the condition that the probability of

Seven-dimensional forest fires

\omega_{xy}>0

One-dimensional long-range diffusion-limited aggregation I

exceeds the threshold for bond percolation on