## Paul Turán Memorial Lectures 2011

This year's Paul Turán Memorial Lectures will be delivered by

Yuval Peres

(Microsoft Research;
Adjunct Professor at The University of
Washington and at UC Berkeley)
- The lectures are organized by the János Bolyai Mathematical Society
and will take place

**in the Main Lecture Hall of the Alfréd Rényi Institute of Mathematics**

### on June 1, 2 and 3 (Wednesday, Thursday, Friday) at 2 pm.

*There will also be an opportunity for young Hungarians to give a talk on a seminar to be chaired by Professor Peres on June 2nd, 15:30 pm.*

*Click here for more details.*.## Schedule of talks:

June 1st, 2pm:

**Laplacian growth**

Abstract: Start with n particles at each of k points in the 2D square lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton. We prove that if the distances between the starting points are suitably scaled, then the set of occupied sites has a deterministic scaling limit. (For k = 1 it is a disk, as proved in 1992 by Lawler, Bramson and Griffeath.) The limiting shape can be described in terms of a free-boundary problem for the Laplacian and quadrature identities for harmonic functions. I will show connections to the rotor-router model, and simulations that suggest intriguing (yet unproved) connections with conformal mapping.

(Joint work with Lionel Levine; J. D'Analyse Math.)

June 2nd, 2pm:

**Mysteries of the abelian sandpile**

Abstract: In the abelian sandpile, particles are added at the origin and whenever a site has four or more particles, the top four particles topple, with one going to each neighbor. Despite similarities to other Laplacian growth models, for the sandpile, the pattern that arises is not circular and depends on the particular lattice. I will show startling empirical similarities between sandpiles in different dimensions, and explain how a principle of least action can be used to bound the spread of the sandpile.

(Joint work with Anne Fey and Lionel Levine; J. Stat Phys.).

June 3rd, 2pm:

**Gravitational allocation to Poisson points**

Abstract: We consider the Poisson fair allocation problem: Given a realization of a Poisson point process, allocate to each point of the process a unit of volume, in a deterministic translation-invariant way, so that the diameter of the region allocated to each point is stochastically as small as possible. One approach to this problem, studied in joint work with C. Hoffman and A. Holroyd, uses the stable marriage algorithm of Gale and Shapley. In dimensions 3 and higher, gravity without inertia yields a satisfying solution. The argument starts with the classical calculation by Chandrasekar of the total force acting on a point, which has a symmetric stable law. The fairness of the allocation is a consequence of the divergence theorem; The diameters of the allocated regions are analyzed using methods from percolation theory.

(Joint work with S. Chatterjee, R. Peled, D. Romik; Annals of Math).