Large deviations, moderate deviations, and the KLS conjecture

Having its origin in theoretical computer science, the Kannan-Lovász-Simonovits (KLS) conjecture is one of the major open problems in asymptotic convex geometry and high-dimensional probability theory today. In this work, we establish a connection between this conjecture and the study of large and moderate deviations for isotropic log-concave random vectors. We then study the moderate deviations for the Euclidean norm of random orthogonally projected random vectors in an ℓpn–ball. This leads to a number of interesting observations: (A) the ℓ1n–ball is critical for the new approach; (B) for p≥2 the rate function in the moderate deviations principle undergoes a phase transition, depending on whether the scaling is below the square-root of the subspace dimensions or comparable; (C) for 1≤p<2 and comparable subspace dimensions, the rate function again displays a phase transition depending on its growth relative to np/2.