Convexity Seminar

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Differentiability of shape functions and effective Lagrangians

Speaker: Yuri Bakhtin, NYU

Location: Warren Weaver Hall 1302

Date: Tuesday, March 31, 2026, 11 a.m.

Synopsis:

In random environments, the cost/energy of an optimal path between two points grows asymptotically linearly with the distance between those points. The function giving the dependence of the deterministic growth rate on direction is called the shape function. Shape functions are always convex and are conjectured to have no corners or flat edges for a broad class of models. This is related to the KPZ universality. For several classes of models (continuous space polymer models at positive and zero temperatures, HJB equations in dynamic random environments, and anisotropic continuous space FPP models), we show that the shape function is differentiable (i.e., has no corners) and give a formula for its gradient. Joint work with Douglas Dow.