## Final Report Summary - SPTRF (Studies in Probability Theory and Related Fields)

Probability theory is among the most vibrant areas of research in pure mathematics today, and is closely tied to many other areas of mathematics and theoretical physics, such as analysis, combinatorics, statistical mechanics and computer science, as well as having applications in virtually all mathematical fields. This project pursues several independent studies in probability theory with a goal of deepening and broadening the connections between probability theory and other fields such as statistical physics, computer science, combinatorics and approximation theory. Selected highlights of the progress made are detailed below.

Significant progress was made in the understanding of the anti-ferromagnetic 3-state Potts model in high dimensions. In joint work with Ohad Feldheim the rigidity phenomenon of the model was established in the setting of periodic boundary conditions. This necessitated the introduction of ideas from algebraic topology which we adapt to the lattice setting. In a second work of Feldheim with Yinon Spinka, a Ph.D. student of the PI, a first proof of the rigidity of the model at low positive temperature is given, establishing the 1985 Kotecky's conjecture.

The project provided understanding of several other models involving hard-core constraints. In joint work with Piotr Milos we considered random surface models in two dimensions, including especially the case of uniformly-sampled Lipschitz functions on the lattice. Adapting a method of Richthammer, our work establishes delocalization of such random surfaces, answering a question mentioned by Brascamp, Lieb and Lebowitz in 1975.

In joint work with Hugo Duminil-Copin, Wojciech Samotij and Yinon Spinka, we study the two-dimensional loop O(n) model. We prove exponential decay of loop lengths when n is large, in analogy with a prediction made by Polyakov in 1975 for the spin O(n) model.

The connections with combinatorics were emphasized in joint work with Greg Kuperberg and Shachar Lovett where we show the existence of regular combinatorial objects which previously were not known to exist, including orthogonal arrays, t-designs, and t-wise permutations with optimal size up to polynomial overhead. The proof is probabilistic and further provides rather precise estimates on the number of such objects of a given size. Our work is the first to show that small t-wise permutations exist, and is also the first to show that the necessary conditions for the existence of (simple) t-designs are also sufficient if lambda is large enough, with a quantitative bound on lambda.

The approximation theory side of the project was advanced with Shoni Gilboa. We obtain bounds on the size of Chebyhsev-type quadratures which are sharp up to constants, for any measure on a compact interval which satisfies a doubling condition. This work, relying on a technique of Kane and results of Mastroianni and Totik, subsumes most of the existing results on the topic. It continues a long line of research which started with the 1937 work of Bernstein and further developed by many authors including Bajnok, Geronimus, Kane, Kuijlaars, Rabau and Wagner.

The integration of the PI at Tel Aviv University is proceeding excellently. The PI has received tenure and was promoted to associate professor status. Recently, the PI has won an ERC starting grant from the European Commission as well as an Israeli science foundation grant to support further aspects of his research.

Project website: http://www.math.tau.ac.il/~peledron/

Significant progress was made in the understanding of the anti-ferromagnetic 3-state Potts model in high dimensions. In joint work with Ohad Feldheim the rigidity phenomenon of the model was established in the setting of periodic boundary conditions. This necessitated the introduction of ideas from algebraic topology which we adapt to the lattice setting. In a second work of Feldheim with Yinon Spinka, a Ph.D. student of the PI, a first proof of the rigidity of the model at low positive temperature is given, establishing the 1985 Kotecky's conjecture.

The project provided understanding of several other models involving hard-core constraints. In joint work with Piotr Milos we considered random surface models in two dimensions, including especially the case of uniformly-sampled Lipschitz functions on the lattice. Adapting a method of Richthammer, our work establishes delocalization of such random surfaces, answering a question mentioned by Brascamp, Lieb and Lebowitz in 1975.

In joint work with Hugo Duminil-Copin, Wojciech Samotij and Yinon Spinka, we study the two-dimensional loop O(n) model. We prove exponential decay of loop lengths when n is large, in analogy with a prediction made by Polyakov in 1975 for the spin O(n) model.

The connections with combinatorics were emphasized in joint work with Greg Kuperberg and Shachar Lovett where we show the existence of regular combinatorial objects which previously were not known to exist, including orthogonal arrays, t-designs, and t-wise permutations with optimal size up to polynomial overhead. The proof is probabilistic and further provides rather precise estimates on the number of such objects of a given size. Our work is the first to show that small t-wise permutations exist, and is also the first to show that the necessary conditions for the existence of (simple) t-designs are also sufficient if lambda is large enough, with a quantitative bound on lambda.

The approximation theory side of the project was advanced with Shoni Gilboa. We obtain bounds on the size of Chebyhsev-type quadratures which are sharp up to constants, for any measure on a compact interval which satisfies a doubling condition. This work, relying on a technique of Kane and results of Mastroianni and Totik, subsumes most of the existing results on the topic. It continues a long line of research which started with the 1937 work of Bernstein and further developed by many authors including Bajnok, Geronimus, Kane, Kuijlaars, Rabau and Wagner.

The integration of the PI at Tel Aviv University is proceeding excellently. The PI has received tenure and was promoted to associate professor status. Recently, the PI has won an ERC starting grant from the European Commission as well as an Israeli science foundation grant to support further aspects of his research.

Project website: http://www.math.tau.ac.il/~peledron/