## Periodic Reporting for period 3 - LocalOrder (Localization and Ordering Phenomena in Statistical Physics, Probability Theory and Combinatorics)

Reporting period: 2019-01-01 to 2020-06-30

Mathematical statistical physics has seen spectacular progress in recent years. Existing problems which were previously unattainable were solved, opening a way to approach some of the classical open questions in the field. The proposed research focuses on phenomena of localization and long-range order in physical systems of large size, identifying several fundamental questions lying at the interface of Statistical Physics, Probability Theory and Combinatorics. One circle of questions concerns the fluctuation behavior of random surfaces, with a main goal of the research to establish some of the long-standing universality conjectures for these fluctuations. This study is also tied to the localization features of random operators, such as random Schrodinger operators and band matrices, as well as those of reinforced random walks. A second circle of questions regards long-range order in discrete spin systems including specifically systems with hard constraints such as proper colorings, independent sets and loop models. For such models it is expected that long-range order holds in great generality in high dimensions while critical phenomena may appear in low dimensions. However, existing rigorous results on models of this type are quite limited.

The PI and Yinon Spinka have developed a method for proving long-range order in discrete spin systems in high dimensions. Let us describe a special case of the result: Color a portion of the cubic lattice Z^d with q colors with the restriction that adjacent vertices receive different colors. How does a typical coloring of this type look like? We have shown that if d is sufficiently large compared with q, most such colorings are very structured. Indeed, in a typical coloring the q colors split into two subsets of equal size (near equal if q is odd) and then most even vertices of the lattice are assigned colors from the first subset while most odd vertices are assigned colors from the second subset. This coloring model is motivated by physics where it is called the zero-temperature limit of the anti-ferromagnetic Potts model. In this context, the new results address questions going back to Berker--Kadanoff (1980), Koteck\'y (1985) and Salas--Sokal (1997). They confirm physicists' prediction of the existence of a broken sub-lattice symmetry (BSS) phase in the anti-ferromagnetic Potts model and further show that such phases constitute a universal phenomenon in discrete spin systems in high dimensions.

A different type of model with hard constraints was considered in work of the PI with Duminil-Copin, Glazman and Spinka. Our work considers the loop O(n) model on the hexagonal lattice - a model of random non-intersecting loops on the lattice having two parameters, an edge weight x and a loop weight n. The model was introduced by physicists Domany, Mukamel, Nienhuis and Schwimmer in 1981 and its phase diagram predicted by Nienhuis in 1982. In our work we prove the existence of macroscopic loops in the model on the critical line predicted by Nienhuis. This is the first instance where the existence of macroscopic loops has been rigorously verified in a loop O(n) model. The results have bearing also on the problem of delocalization of integer-valued random surfaces, where they join a very limited set of previously discovered cases where such a surface is known to delocalize.

A different type of model with hard constraints was considered in work of the PI with Duminil-Copin, Glazman and Spinka. Our work considers the loop O(n) model on the hexagonal lattice - a model of random non-intersecting loops on the lattice having two parameters, an edge weight x and a loop weight n. The model was introduced by physicists Domany, Mukamel, Nienhuis and Schwimmer in 1981 and its phase diagram predicted by Nienhuis in 1982. In our work we prove the existence of macroscopic loops in the model on the critical line predicted by Nienhuis. This is the first instance where the existence of macroscopic loops has been rigorously verified in a loop O(n) model. The results have bearing also on the problem of delocalization of integer-valued random surfaces, where they join a very limited set of previously discovered cases where such a surface is known to delocalize.

In the second half of the project the PI plans to continue the research on the theory of random surfaces, random operators and the O(n) model, as well as to complete the research projects suggested in the ERC project proposal. Preliminary progress has already been made towards problems of localization of random surfaces in dimensions d>=3 for a general class of interaction potentials, for the delocalization of certain integer-valued height functions satisfying a Lipschitz constraint and for establishing macroscopic loops in a larger portion of the predicted phase diagram of the loop O(n) model. These advances required the development of new tools and methodology and it is hoped that these in turn will lead to further progress on the core problems of the proposal.