Jacopo

18.604 – Seminar in Probability (Random walks and uniform spanning trees) | Fall 2025

Teaching

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 »

18.604 – Seminar in Probability (Random walks and uniform spanning trees) | Spring 2025

Teaching

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, …

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Directed distance in bipolar-oriented triangulations: A path toward the directed Liouville quantum gravity metrics

Talks

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 …

Directed distance in bipolar-oriented triangulations: A path toward the directed Liouville quantum gravity metricsRead More »

The longest increasing subsequence of Brownian separable permutons

Talks

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 …

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[24] Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits (with Ewain Gwynne).

Preprints

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 …

[24] Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits (with Ewain Gwynne).Read More »

[23] Permuton and local limits for the Luce model (with Sourav Chatterjee and Persi Diaconis).

Preprints

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 …

[23] Permuton and local limits for the Luce model (with Sourav Chatterjee and Persi Diaconis).Read More »

[22] Surface sums in two-dimensional large-N lattice Yang–Mills: Cancellations and explicit computation for general loops (with Sky Cao and Jasper Shogren-Knaak).

Preprints

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 …

[22] Surface sums in two-dimensional large-N lattice Yang–Mills: Cancellations and explicit computation for general loops (with Sky Cao and Jasper Shogren-Knaak).Read More »

[21] The longest increasing subsequence of Brownian separable permutons (with Arka Adhikari, Thomas Budzinski, William Da Silva, Delphin Sénizergues).

Preprints

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 …

[21] The longest increasing subsequence of Brownian separable permutons (with Arka Adhikari, Thomas Budzinski, William Da Silva, Delphin Sénizergues).Read More »

High-dimensional permutons: the Schnyder wood and Brownian separable d-permuton

Images

The simulations in this page are for this work on the high-dimensional theory for permutons.  The Schnyder wood permuton & the Brownian separable 3-permuton Simulations for two 3-dimensional permutons. Each simulation is spinning along the Z-axis (w.r.t. the notation used in our paper). The 3-dimensional permuton associated with a permutation of size 10000 sampled from the Schnyder …

High-dimensional permutons: the Schnyder wood and Brownian separable d-permutonRead More »

[20] High-dimensional permutons: theory and applications (with Andrew Lin).

Preprints

Permutons, which are probability measures on the unit square with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a -dimensional generalization of these measures for all , which we call –dimensional permutons, and extend — from the two-dimensional setting — the theory to prove convergence of sequences of (random) …

[20] High-dimensional permutons: theory and applications (with Andrew Lin).Read More »