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Probability and Combinatorics

Tuesday, April 25, 2017 - 3:00pm

Christopher Sinclair

University of Orgeon


Temple University

Wachman Hall 617

Note the room change.

We consider the distribution of N p-adic particles with interaction energy proportional to the log of the p-adic distance between two particles. When the particles are constrained to the ring of integers of a local field, the distribution of particles is proportional to a power of the p-adic absolute value of the Vandermonde determinant.  This leads to a first question: What is the normalization constant necessary to make this a probability measure?  This sounds like a triviality, but this normalization constant as a function of extrinsic variables (like number of particles, or temperature) holds much information about the statistics of the particles.  Viewed another way, this normalization constant is a p-adic analog of the now famous Selberg integral. While a closed form for this seems out of reach, I will derive a remarkable identity that may hold the key to unlocking more nuanced information about p-adic electrostatics.  Along the way we’ll find an identity for the generating function of probabilities that a degree N polynomial with p-adic integer coefficients split completely.  Joint work with Jeff Vaaler.