## Departmental PhD Thesis Exam – Alexandru Gatea

Monday, September 12, 2022 at 3:00 p.m. (sharp)

PhD Candidate: Alexandru Gatea
Supervisor: Balint Virag
Thesis title:  Grid entropy in last passage percolation, a variational formula for Gibbs Free Energy, and applications to a “choose the best of D samples” model

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Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we develop the notion of grid entropy: a deterministic directed norm with negative sign that measures the proportion of empirical measures of edge weights (in a fixed direction or direction-free) which converge weakly to a given target
measure. We study various properties of grid entropy, including an upper bound on the sum of relative and grid entropies, upper semicontinuity in most cases, and the fact that grid entropy can be described as the negative convex conjugate of Gibbs Free Energy. We show that the direction-free case is nothing more than the direction-fixed case in the (1, 1, . . . , 1) direction. In addition, we derive a grid entropy variational formula for the point-to-point/point-to-hyperplane Gibbs Free Energies that answers a directed polymer version of a question of Hoffman. Shifting gears, we proceed to study the limiting behaviour of empirical measures in a model consisting of repeatedly taking D samples from a distribution and picking out one according to an omniscient “strategy.” We show that the set of limit points of empirical measures is almost surely the same whether or not we restrict ourselves to strategies which make the choices independently of all
past and future choices, and furthermore, that this set coincides with the set of measures with finite grid entropy. These sets are convex and weakly compact; we characterize their extreme points as those given by a natural “greedy” deterministic strategy and we compute the grid entropy of said extreme points to be 0. This yields a description of the set of limit points
of empirical measures as the closed convex hull of measures given by a density which is D ¨ Beta(1, D) distributed. We also derive a simplified version of a grid entropy-based variational formula for Gibbs Free Energy for this model, and we present the dual formula for grid entropy.

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A copy of the thesis can be found here: thesis