Description: I will introduce a new universal family of random permutons, called the skew Brownian permutons, describing the scaling limit of various natural models of random constrained permutations. After that, the main goal will be to discuss some connections between random permutations and random geometry. In particular, we will focus on the problem of the longest …
Lattice Yang-Mills theories are important models in particle physics. They are defined on the d-dimensional lattice using a group of matrices of dimension , and Wilson loop expectations are the fundamental observables of these theories. Recently, Cao, Park, and Sheffield showed that Wilson loop expectations can be expressed as sums over certain embedded bipartite maps of any genus. Building on this novel approach, …
Lattice Yang-Mills theory in the large N limit via random surfacesRead More »
Description: In 1912 Henri Poincaré asked the following simple question: “In how many different ways can a simple loop in the plane, called a meander, cross a line a specified number of times?” Despite many efforts, this question remains open after over a century. In the first part of the mini-course, I will present the …
We obtain scaling and local limit results for large random multirectangular Young tableaux via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). In particular, we find an explicit description of the limiting surface, based on solving a complex-valued polynomial equation. As a consequence, we find a simple criteria to …
This is the webpage for the course of Math 159 – Discrete Probabilistic Methods. Lecture times: Tuesday and Thursday, 1:30 PM – 2:50 PM (Hewlett Teaching Center, Rm 101). Instructor: Jacopo Borga, jborga_at_stanford.edu. Office hours: Tuesday 11:00 AM-12:15 PM (Building 380, 382-Q2). Assistant: / Office hours: / Description: In this course we will cover a range of different topics …
MATH 159 – Discrete Probabilistic Methods | 2023-2024Read More »
The images on this page are taken from this work on the scaling and local limits of random Young tableaux. The heart example Figures for the heart example. Top-left: The Young diagram considered in the heart example. Top-right: The frozen boundary curve of the corresponding liquid region. Bottom (from left to right): a uniform random tableau …
Scaling and local limits of random Young tableauxRead More »
We obtain scaling and local limit results for large random Young tableaux of fixed shape via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). More precisely, we prove: an explicit description of the limiting surface of a uniform random Young tableau of shape , based on solving a complex-valued …
A (rather informal) presentation of the results related to the scaling limit of Baxter permutations can be found here.
The images on this page are taken from this work on meanders and this work on meandric systems. Meanders and meandric permutations Left: Two large uniform meanders of size 256 and 2048. Right: The plots of the two corresponding meandric permutations. These simulations are obtained using this paper‘s Markov chain Monte Carlo algorithm. Meandric systems Simulation of a uniform …
Meanders, meandric permutations, and meandric systemsRead More »
What is the behavior of the longest increasing subsequence of a uniformly random permutation? Its length is of order plus Tracy–Widom fluctuations of order . Its scaling limit is the directed geodesic of the directed landscape. This talk discusses how this behavior changes dramatically when one looks at universal Brownian-type permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants such that …
Long increasing subsequences in Brownian-type permutationsRead More »