Statistical Mechanics & Complex Zeros: it's a match!

Localizing the complex zeros of certain large degree polynomials is crucial in Statistical Mechanics in order to identify the phase transitions of the system. I will introduce this beautiful bridge across disciplines mainly through two examples: the matching model and the Ising model on graphs. The meaning of "phase transition" and its connection with complex zeros will be explained. Then I will show how to localize the zeros of the matching polynomials in order to prove the absence of phase transitions in matching models (Heilmann-Lieb). In the end I will mention that a similar approach is succesfull to localize the phase transitions in Ising models (Lee-Yang).