Séminaire OrléansZeros of the Ising partition function
Han Peters (Amsterdam)
Thursday 19 April 2018 14:00 - Orléans - Salle de Séminaire
Work in progress with Guus Regts The Ising model originated in statistical physics, where it was used to model magnetic objects. Absence of zeros of the partition function implies the absence of phase transitions, i.e. it implies analytic dependence on the parameters, a classical result of Lee and Yang. More recently there has been considerable interest from the perspective of algorithmic complexity. In complexity theory, absence of zeros implies the existence of fast approximation algorithms. The partition function of a graph depends on two parameters, a field-like parameter $\lambda$, and a temperature-like parameter $\beta$. A result of Lee-Yang from 1952 states that for the physically relevant $\beta \in [-1,1]$, the partition function can only be zero when $\lambda$ lies on the unit circle. The famous Lee-Yang Theorem has received enormous attention in the literature. The exact $\lambda$-values for which the partition function is always non-zero has been studied, but not completely described. We will give a complete description for arbitrary graphs of maximal degree $d \ge 2$. Our result is obtained by reducing the problem to the setting of trees, where it can be studied using classical techniques from complex dynamical systems.