My research lies broadly in probability and its interactions with statistical physics, random matrix theory, free probability, optimal transport, representation theory and convex geometry. A recurring theme in this work is the correspondence associating a vector in Euclidean space with its empirical coordinate measure.
S. G. G. Johnston (2025).
On creating convexity in high dimensions
We study a version of a 1990s conjecture of Talagrand on whether repeated Minkowski sums of a large subset of Euclidean space must contain a large convex subset. Using empirical coordinate measures, we reformulate the problem in optimal transport and prove a negative result via Monge–Kantorovich duality.
Talagrand’s original conjecture has since been proved by Hua, Song and Tudose.
O. Arizmendi and S. G. G. Johnston (2026).
Free probability via entropic optimal transport.
Invent. Math. (to appear)
Marcus, Spielman and Srivastava proved that the expected characteristic polynomial of a sum of two randomly conjugated diagonal matrices is the same whether the conjugation is by a Haar unitary or a uniform permutation matrix. We study this identity in the limiting regime where the empirical spectra of the two matrices converge to given probability measures. Using large deviation techniques, we show that free convolution, along with several other operations in free probability, admits an optimal transport formulation.
S. G. G. Johnston and J. Prochno (2024).
The macroscopic shape of Gelfand–Tsetlin patterns and free probability.
We prove a large deviation principle for large random Gelfand–Tsetlin patterns, describing the asymptotic empirical measure of the interior rows. In particular, this proves a conjecture of Shlyakhtenko and Tao that the Euler–Lagrange equations for free compression arise from the statistical mechanics of interlacing patterns.
S. G. G. Johnston (2025).
Continuous Kasteleyn theory for the bead model.
J. Eur. Math. Soc.
In the 1960s, Kasteleyn realised that partition functions for various models in statistical mechanics, such as domino tilings of a chessboard, can be written in terms of determinants of certain operators. In this article we develop a continuous version of Kasteleyn theory, showing that partition functions for interlacing patterns on the torus can be expressed as Fredholm determinants. The explicit formulas obtained here are used in the article above to study the macroscopic shape of Gelfand–Tsetlin patterns.
