## Probability theory

Probability theory is among the most important areas of research in mathematics. It is crucial to many other disciplines including analysis, combinatorics, statistical mechanics and computer science.

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The SPTRF (Studies in probability theory and related fields) project has pursued several interrelated studies in probability theory, with the goal of deepening and broadening the connections between probability theory and other fields.

Significant progress has been made in understanding the anti-ferromagnetic 3-state Potts model in high dimensions. The rigidity phenomenon of the model was established in the setting of periodic boundary conditions. This necessitated the introduction of ideas from algebraic topology that were adapted to the lattice setting. A first proof of the rigidity of the model at low positive temperature was developed, establishing the 1985 Kotecky's conjecture.

The project has also provided an understanding of other models involving hard-core constraints. Random surface models in two dimensions were considered, including the case of uniformly sampled Lipschitz functions on the lattice. The work establishes delocalisation of such random surfaces. The two-dimensional loop O(n) model was investigated. The exponential decay of loop lengths was demonstrated when n is large.

The connections with combinatorics were emphasised in another collaboration. They demonstrated the existence of new regular combinatorial objects including orthogonal arrays, t-designs, and t-wise permutations with optimal size up to the polynomial overhead. The proof is probabilistic and provides further estimates of the number of such objects of a given size.

This work is the first to show that small t-wise permutations exist. These permutations determine the necessary conditions for the existence of (simple) t-designs, if lambda is large enough, with a quantitative bound on lambda.

Approximation theory was also advanced. Bounds were obtained on the size of Chebyhsev-type quadratures, which are sharp up to constants, for any measure on a compact interval that satisfies a doubling condition. This work unifies many existing results on the topic.

Significant progress has been made in understanding the anti-ferromagnetic 3-state Potts model in high dimensions. The rigidity phenomenon of the model was established in the setting of periodic boundary conditions. This necessitated the introduction of ideas from algebraic topology that were adapted to the lattice setting. A first proof of the rigidity of the model at low positive temperature was developed, establishing the 1985 Kotecky's conjecture.

The project has also provided an understanding of other models involving hard-core constraints. Random surface models in two dimensions were considered, including the case of uniformly sampled Lipschitz functions on the lattice. The work establishes delocalisation of such random surfaces. The two-dimensional loop O(n) model was investigated. The exponential decay of loop lengths was demonstrated when n is large.

The connections with combinatorics were emphasised in another collaboration. They demonstrated the existence of new regular combinatorial objects including orthogonal arrays, t-designs, and t-wise permutations with optimal size up to the polynomial overhead. The proof is probabilistic and provides further estimates of the number of such objects of a given size.

This work is the first to show that small t-wise permutations exist. These permutations determine the necessary conditions for the existence of (simple) t-designs, if lambda is large enough, with a quantitative bound on lambda.

Approximation theory was also advanced. Bounds were obtained on the size of Chebyhsev-type quadratures, which are sharp up to constants, for any measure on a compact interval that satisfies a doubling condition. This work unifies many existing results on the topic.