##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability Seminar

## Lionel Levine

#### Cornell University

## Threshold state of the abelian sandpile

##### Abstract:

A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile $s_0$ if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a "threshold state'' $s_T$ that topples forever. Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in $s_T$ in the limit as $s_0$ tends to negative infinity. I will outline how this conjecture was proved by means of a Markov renewal theorem.

Host: Todd Kemp

### November 12, 2015

### 9:00 AM

### AP&M 6402

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