Abstract:

In this talk we review a number of old results concerning certain statistical mechanics models and their possible connections to the Riemann Hypothesis. 

A standard reformulation of the Riemann Hypothesis (RH) is: The (two-sided) Laplace transform of a certain specific function \Psi on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also \Psi is a positive integrable function, so (modulo a multiplicative constant C) is a probability density function.  

A (finite) Ising model is a specific type of probability measure P on the points S=(S_1,...,S_N) with each S_j = +1 or -1. The Lee-Yang theorem (of T. D. Lee and C. N. Yang) implies that that for non-negative a_1, ..., a_N, the Laplace transform of the induced probability distribution of a_1 S_1 + ... + a_N S_N has only pure imaginary zeros.  

The big question here is whether it's possible to find a sequence of Ising models so that the limit as N tends to \infty of such distributions has density exactly C \Psi. We'll discuss some hints as to how one might try to do this.