The Tutte polynomial and the Potts model in square lattices
by Arun Mani
Abstract: The Tutte polynomial of a graph is a polynomial in two
variables, $x,y$, that is of central importance in many counting problems. The Potts model partition function is known to be computationally equivalent to an evaluation of the Tutte polynomial
along the curve $(x-1)(y-1) = q$ for integer values of $q$. The asymptotic growth of the Tutte polynomial (and the Potts model) of a square lattice as its dimensions tend to infinity are often of special interest in combinatorics and statistical physics. In this talk we
introduce a family of inequalities for the Tutte polynomial of square lattices, and apply them to obtain non-trivial one-sided bounds for a limit describing this growth when $x, y \geq 1$.
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