Permutations in Random Geometry

Stanford University, Stanford CA

Investigators

Abstract

This project lies at the intersection of probability theory, combinatorics, and mathematical physics. Its primary objective is to uncover novel connections between two currently active research domains that have developed independently until recently: random permutations and random geometry. The emerging interplay between permutons (limits of random permutations) and random geometric objects arising in quantum physics and statistical mechanics (such as Schramm–Loewner evolution curves and Liouville quantum gravity surfaces) will play a fundamental role in generating significant advancements in both fields. This will involve formulating novel theories for universal random permutons and random directed metrics, expanding existing ones, and effectively resolving long-standing problems on meanders and meandric systems. The three main objectives of this research project are, first, to investigate the problem of the longest increasing subsequence of random permutations from a novel angle, which involves linking it to directed metrics in planar maps. The goal is to construct a 'quantum version' of the universal Kardar-Parisi-Zhang geometry, i.e., the directed landscape. Second, to study the geometry of random meanders and broader statistical physics models involving crossing fully packed loops on planar maps. The objective is to tackle the long-standing open problem of determining the scaling limit of random uniform meanders and meandric permutations. Third to establish connections between the limits of d-dimensional permutations and new scale-invariant d-dimensional random geometries introduced in the physical literature. The aim is to begin developing a novel d-dimensional theory of random geometries and permutons. The project offers opportunities for education and outreach to high school and undergraduate students, as well as mentoring of undergraduate and Ph.D. students. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.