  # Localization and Ordering Phenomena in Statistical Physics, Probability Theory and Combinatorics

## Periodic Reporting for period 4 - LocalOrder (Localization and Ordering Phenomena in Statistical Physics, Probability Theory and Combinatorics)

Reporting period: 2020-07-01 to 2021-12-31

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. The project successfully developed new tools and mathematical theorems to address the challenges that these questions pose and advance the state-of-the-art in mathematical statistical physics.
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), Kotecký (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. In our work we prove the existence of macroscopic loops in the model on the critical line predicted by Nienhuis (1982). 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.

The PI with Michael Aizenman, Jeffrey Schenker, Mira Shamis and Sasha Sodin developed new tools to study random matrices and the Wegner orbital model.

The PI with Nishant Chandgotia, Martin Tassy and Scott Sheffield established the delocalization of the height function of square ice (random graph homomorphisms from Z^2 to Z). The proof introduces a new method for showing delocalization based on the earlier work of Sheffield combined with an argument which capitalizes on the local instability of the height function - being forced to take even values at even vertices and odd values at odd vertices.

The PI and Gady Kozma proved power-law decay for the weights in the two-dimensional vertex-reinforced jump process (VRJP). The result enters as an ingredient to the framework developed by Sabot and Zeng which then yields that the VRJP is recurrent in two dimensions.

Alexander Glazman and Ioan Manolescu proved the existence of macroscopic level lines for uniformly-sampled Lipschitz functions on the triangular lattice (the point n=2, x=1 in the phase diagram of the loop O(n) model).

The PI with Nicholas Crawford, Alexander Glazman and Matan Harel proved the existence of macroscopic loops in the loop O(n) model in a region near the critical percolation point n=x=1. This work is the first to establish the existence of macroscopic loops in a region of positive Lebesgue measure of the phase diagram, supporting the 1982 predictions of Nienhuis on the model. The work introduces a new method for proving delocalization, based on a "XOR trick" and a result concerning the site-percolation threshold of circle-packing graphs.

The PI and Alexander Magazinov proved new concentration inequalities of Brascamp-Lieb type for log-concave distributions. These concentration inequalities were then applied to random surfaces with grad-phi interactions to prove their localization in dimensions d>=3 (with further control of their fluctuations in two dimensions) and to control their tail behavior. The results address questions going back to Brascamp-Lieb-Lebowitz (1975), Deuschel-Giacomin (2000) and Velenik (2006).

The PI gave invited lectures and mini-courses on the topics of the grant at many venues including online seminars, conference presentations and mini-courses.
Progress was made beyond the state-of-the-art and expected results of the project.
The PI and Alexander Glazman investigated the 6-vertex model in its anti-ferroelectric regime as well as along the boundary between the anti-ferroelectric and disordered regimes. The results cover the 6-vertex representation as well as the associated height function and spin representations. Further results are deduced for the Ashkin-Teller model along its critical line by presenting a coupling between the two models.

The PI obtained a universal lower bound for the site percolation threshold of certain planar graphs, obtained via the theory of circle packings. The result makes progress towards conjectures of Benjamini (2018) for percolation on general planar graphs and further forms an essential ingredient for the progress described above on the loop O(n) model.

The PI with Gady Kozma, Tom Meyerovitch and Wojciech Samotij considered a random metric space sampled from the metric polytope (i.e. a random metric space obtained by choosing uniformly, via Lebesgue measure, among all metric spaces with bounded diameter). The work develops a new approach to the problem via entropy techniques and obtains a precise description of the typical properties of the random metric space as well as precise estimates for the volume of the metric polytope itself.

The PI with Michael Aizenman, Matan Harel and Jacob Shapiro developed a new approach to the Berezinskii-Kosterlitz-Thouless transition in plane-rotator models. The approach builds a direct connection between slow decay of correlations in the rotator model and the delocalization of the dual integer-valued height function. The picture is completed by introducing new correlation inequalities and relying on a recent general proof of delocalization in integer-valued height functions by Lammers.