In this paper we modify the method of Nazarov, Peres, and Volberg [14] to get an estimate from above of the Buffon needle probability of the nth partially constructed Sierpinski gasket of Hausdorff dimension 1.

This paper considers the p-Laplacian PDE for p = 1,∞ and some interesting new game theoretic interpretations, due to Kohn–Serfaty [K-S] and to Peres–Schramm–Sheffield–Wilson [P-S-S-W].

In this paper we modify the method of Nazarov, Peres, and Volberg [14] to get an estimate from above of the Buffon needle probability of the nth partially constructed Sierpinski gasket of Hausdorff dimension 1.

We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on ...

............................................................................... i Grant Information.................................................................... ii Dedication............................................................................. iii Acknowledgement.................................................................... iv List of Figures...................................

We study the rotor router model and two deterministic sandpile models. For the rotor router model in Z, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter h (the initial number of particles at each site), allows to prove a number of new limitin...