In recent years, surprising connections have emerged between the theory of random permutations and random geometric objects arising in probability, statistical mechanics, and quantum physics. On one side are permutons, which describe scaling limits of large permutations; on the other are random planar maps, Schramm–Loewner evolution curves, and Liouville quantum gravity surfaces. In this talk, I will discuss some of these …
We study a natural one-parameter family of random bipolar-oriented planar maps which lies in the -Liouville quantum gravity (LQG) universality class for . For these maps, we identify exact scaling exponents for directed graph distances. Writing for the size of the map, the longest directed paths have lengths comparable to (up to constants), while the …
Seminar in Probability Random walks and uniform spanning trees Course 18.604, Fall 2025 Professor: Jacopo Borga (jborga@mit.edu) Office hours: I am available to help you with your presentations (more details below) – email me about it! For questions about exercises in the homework please ask our course assistant Alexis Zhou (leqizhou@mit.edu) during her office hours or send an …
18.604 – Seminar in Probability (Random walks and uniform spanning trees) | Fall 2025Read More »
Seminar in Probability Random walks and uniform spanning trees Course 18.604, Spring 2025 Professor: Jacopo BorgaLinks to an external site. (jborga@mit.edu) Communication specialists: Mary Caulfield (mcaulf@mit.edu) and Susan Ruff (ruff.susan@gmail.com) Class Schedule: TR 9.30 AM – 11AM (2-151) & CalendarLinks to an external site.. Attendance and active participation in all classes are mandatory and will be enforced. If you are unable to attend a class, …
18.604 – Seminar in Probability (Random walks and uniform spanning trees) | Spring 2025Read More »
Last and first passage percolation in two dimensions are classical discrete models of random directed planar Euclidean metrics in the KPZ universality class. Their scaling limit is described by the directed landscape of Dauvergne-Ortmann-Virág. Random planar maps are classical discrete models of random undirected planar fractal metrics in the LQG universality class. Their scaling limit is described by the (undirected) LQG metric of Ding-Dubédat-Dunlap-Falconet …
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 …
The longest increasing subsequence of Brownian separable permutonsRead More »
We study longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge. We construct the Busemann function which measures directed distance to along a natural interface in the UIBOT. We show …
We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the …
In the context of two-dimensional large- lattice Yang–Mills theory, we perform a refined study of the surface sums defined in the companion work [19]. In this setting, the surface sums are a priori expected to exhibit significant simplifications, because two-dimensional Yang–Mills theory is a special model admitting many known exact formulas. Thus, a natural problem is …
We establish a scaling limit result for the length LIS of the longest increasing subsequence of a permutation of size sampled from the Brownian separable permuton of parameter , which is the universal limit of pattern-avoiding permutations. Specifically, we prove that where is the unique solution in the interval to the equation and is a non-deterministic …