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.
