Student Probability/PDE Seminar: Geomety of Polymers in Last Passage Percolation Under Various Constraints

Seminar | September 1 | 2:10-3:30 p.m. | 891 Evans Hall

 Shirshendu Ganguly, UC Berkeley

 Department of Mathematics

Motivated by extremal isoperimetric problems in percolation, I shall describe a model which puts a global curvature constraint on the standard Last passage percolation model and studies the longest increasing path from $(0, 0)$ to $(n, n)$ trapping atypically large area. As is typical in these models, the first order behaviour of this random contour is determined by a variational problem which we explicitly solve. More interesting are exponents related to local fluctuation properties which capture the competition between the global curvature constraint and the behaviour of an unconstrained path governed by KPZ universality. These can be studied via maximal facet lengths of the convex hull of the contour and the maximal inward deviation from the hull for which we identify scaling exponents $3/4$ and $1/2$ respectively. If time permits, I will also describe certain large deviation problems, concerning the behavior of the geodesic under other natural constraints.