# Master2019 Proposal

### From Werner KRAUTH

**Title: An improved bounding density for the liquid phase in hard-sphere systems.**

**Stage de M2 2019, ENS (Werner Krauth)**

**Summary**

While the existence of a paramagnetic-ferromagnetic phase transition in the two-dimensional Ising model was proven some 80 years ago (Peierls, 1936), the rigorous proof of a phase transition in the hard-disk model (and more generally in hard-sphere models in dimension d>1) is still missing (the phase diagram of the hard-disk model was obtained in 2011. The same is true for general particle models. In fact, the most recent rigorous result for hard disks is due to Lebowitz and Penrose, in 1964: It states that the liquid phase in this model exists. It extends at least from density =0 to density = 0.03619. (This is to be compared to the empirical value 069).)

In 2003, however, Kannan, Mahoney, and Montenegro proved that simple Monte Carlo algorithms showed the property of `fast' mixing up to much larger densities (density = 1/8= 0.125), more than three times larger than those indicated by Lebowitz and Penrose. Fast mixing appears incompatible with any presence of order or slower than exponential correlation functions. The task of this practical is to rigorously prove this conjecture.

The "stage" will be located at LPENS, and supervised by Werner Krauth. A colloboration with the mathematicians C. Moore (Santa Fe Institute, USA) and T. Hayes (University of New Mexico, USA) appears likely. Moore and Hayes recently proved an improved rapid-mixing bound, without obtaining a connection between the mixing times and the physical properties of the system.