Jacopo

MATH 159 – Discrete Probabilistic Methods | 2023-2024

Teaching

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 …

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[17] A determinantal point process approach to scaling and local limits of random Young tableaux (with Cédric Boutillier, Valentin Féray and Pierre-Loïc Méliot).

Preprints

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 …

[17] A determinantal point process approach to scaling and local limits of random Young tableaux (with Cédric Boutillier, Valentin Féray and Pierre-Loïc Méliot).Read More »

Meanders, meandric permutations, and meandric systems

Images

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 …

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Long increasing subsequences in Brownian-type permutations

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 Brownian separable permutons. We show that there are explicit constants such that …

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[16] Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons (with William Da Silva and Ewain Gwynne).

Preprints

The Brownian separable permutons are a one-parameter family – indexed by – of universal limits of random constrained permutations. We show that for each , there are explicit constants such that the length of the longest increasing subsequence in a random permutation of size sampled from the Brownian separable permuton is between and with probability …

[16] Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons (with William Da Silva and Ewain Gwynne).Read More »

Meanders and Meandric Systems

Talks

In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?” Despite many efforts, this question remains very open after more than a century. In this talk, I will present the conjectural scaling limit …

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MATH 151 – Introduction to Probability | 2022-2023

Teaching

This is the webpage for the course of Math 151 – Introduction to Probability. Lecture times: Tuesday and Thursday 9:00 AM – 10:20 AM (Room: 380-380X). Instructor: Jacopo Borga, jborga_at_stanford.edu. Office hours: Thursday 11:00-12:15 AM (Building 380, Room 382-Q2). Assistant: Zhihan Li, zhihanli@stanford.edu.  Office hours: Mondays and Fridays 11:30 AM-1:30 PM (Building 380, Room 381A). Description: This is a first …

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