Colloquium Mathematics - Dr. E.A. Verbitskiy, University of Leiden

Title: DECIMATION LIMITS OF PRINCIPAL ALGEBRAIC Zd-ACTIONS

Abstract:

In this expository talk, using simple examples, we will discuss renormalization of certain algebraic dynamical systems. In joint work with E.Arzhakova, K.Schmidt, and D.Lind, we prove that by suitably renormalizing these decimations we can identify a limiting behavior given by a continuous concave function on the Newtonpolytopeof f, and show that this decimation limit is the negative of the Legendre dual of the Ronkin function of f. In certain cases, the decimation limit coincides with the surface tension of random surfacesrelated to dimer models, but the statistical physics methods used to prove this are quite different. The proof is based on Mahler’s estimates on polynomialcoefficients using Mahler measure, and an idea used by Boyd to prove that Mahler measure is continuous in the coefficients of the polynomial. Refinements of convergence questions involve diophantine issues that I will discuss, together with some open problems.