The large and moderate deviations approach in geometric functional analysis

Abstract: The work of Gantert, Kim, and Ramanan [Large deviations for random projections of $\ell^p$ balls, Ann. Probab. 45 (6B), 2017] has initiated and inspired a new direction of research in the asymptotic theory of geometric functional analysis. The moderate deviations perspective, describing the asymptotic behavior between the scale of a central limit theorem and a large deviations principle, was later added by Kabluchko, Prochno, and Th\"ale in [High-dimensional limit theorems for random vectors in $\ell_p^n$ balls. II, Commun. Contemp. Math. 23(3), 2021]. These two approaches nicely complement the classical study of central limit phenomena or non-asymptotic concentration bounds for high-dimensional random geometric quantities. Beyond studying large and moderate deviations principles for random geometric quantities that appear in geometric functional analysis, other ideas emerged from the theory of large deviations and the closely related field of statistical mechanics, and have provided new insight and become the origin for new developments. Within less than a decade, a variety of results have appeared and formed this direction of research. Recently, a connection to the famous Kannan-Lov\'asz-Simonovits conjecture and the study of moderate and large deviations for isotropic log-concave random vectors was discovered. In this manuscript, we introduce the basic principles, survey the work that has been done, and aim to manifest this direction of research, at the same time making it more accessible to a wider community of researchers.