Webpage of Thomas Budzinski

Who am I ?

Since the beginning of 2021, I am a "chargé de recherche" (Junior permanent research position) at the UMPA in the ENS Lyon. My office is MGN1-460. Before that, I have worked in the probability group of the Mathematics Department at UBC (Vancouver), at the mathematics department of the ENS Paris, and in Orsay. My main domain of research is the study of random planar maps, but I am also interested in hyperbolic geometry and all kinds of discrete random objects like random trees, random graphs and random permutations. I am particularly interested in high genus phenomena on random surfaces. Since September 2024, I co-supervise with Grégory Miermont the PhD thesis of Tanguy Lions. Here is my CV.

Research papers

• The longest increasing subsequence of Brownian separable permutons, with Arka Adhikari, Jacopo Borga, William da Silva and Delphin Sénizergues, preprint [slides]
• The critical Karp-Sipser core of Erdös-Rényi random graphs, with Alice Contat, to appear in Annales de l'Institut Henri Poincaré B [Mathematica sheet]
• Local limits of random bipartite maps in high genus: the general case, in Electronic Communications in Probability
• Distances and isoperimetric inequalities in random triangulations of high genus, with Guillaume Chapuy and Baptiste Louf, in Annals of Probability [slides] [video by Guillaume]
• Maximum Agreement Subtrees and Hölder homeomorphisms between Brownian trees, with Delphin Sénizergues, in Journal de l'École Polytechnique [slides] [video]
• The critical Karp-Sipser core of random graphs, with Alice Contat and Nicolas Curien, in Annals of Applied Probability [video by Alice]
• On Cheeger constants of hyperbolic surfaces, with Nicolas Curien and Bram Petri, to appear in Inventiones Mathematicae [slides]
• Multi-ended Markovian triangulations and robust convergence to the UIPT, in Annales Henri Lebesgue [slides]
• Local limits of bipartite maps with prescribed face degrees in high genus, with Baptiste Louf, in Annals of Probability
• Cooperative and stochastic multi-player multi-armed bandits: optimal regret with neither communication nor collisions, with Sébastien Bubeck and Mark Sellke, at COLT 2021
• Coordination without communication: optimal regret in two players multi-armed bandits, with Sébastien Bubeck, at COLT 2020 [slides] [video]
• Recurrence of the Uniform Infinite Half-Plane Map via duality of resistances, with Thomas Lehéricy, in Annals of Probability [slides] [video]
• The diameter of random Belyi surfaces, with Nicolas Curien and Bram Petri, in Algebraic and Geometric Topology
• On the minimal diameter of closed hyperbolic surfaces, with Nicolas Curien and Bram Petri, in Duke Mathematical Journal [slides]
• Universality for random surfaces in unconstrained genus, with Nicolas Curien and Bram Petri, in Electronic Journal of Combinatorics [slides]
• Local limits of uniform triangulations in high genus, with Baptiste Louf, in Inventiones Mathematicae [slides] [video]
• Supercritical causal maps: geodesics and simple random walk, in Electronic Journal of Probability [slides]
• Infinite geodesics in hyperbolic random triangulations, in Annales de l'Institut Henri Poincaré B
• On the mixing time of the flip walk on triangulations of the sphere, in Comptes-rendus mathématiques de l'Académie des Sciences [slides]
• The hyperbolic Brownian plane, in Probability Theory and Related Fields [slides]

Other works (in french)

• Un résultat de recherche obtenu grâce à des lycéens, with Giancarlo Lucchini Arteche, in the Gazette des Mathématiciens.
• My Master Thesis, under the supervision of Nicolas Curien, about the spatial Markov property of planar maps [slides]
• My ENS first year thesis, with Jaouad Mourtada, under the supervision of Harald Helfgott, about the diameter of permutation groups
• My TIPE about percolation, and the slides of the talk I gave about the same topic at the Students seminar of lycée Louis-le-Grand.

Nice pictures

Teaching

• In the first semestre of 2025-2026, I give an M2 course on Random walks at the ENS of Lyon. The lecture notes will be updated as the course progresses. Here are a few references that can be useful to find exercises or to go further.
  • First part of the course: Random walks and graphs by Nicolas Curien.
  • For more on random trees: Arbres et marches aléatoires by Kortchemski (in french, similar to what we have done in class but goes a bit further and gives more context), Random trees and applications by Le Gall (mostly on scaling limits). See also some nice pictures by Igor Kortchemski.
  • Second part: Probability on Trees and Networks by Lyons and Peres (with hints for the exercises in the end), Probability and Geometry on Groups by Pete.
  Here are the partial exam and the final exam from 2024. Here is also the partial exam of 2025 with the grading scheme and some sketches of solutions.
• See here for previous years (in french).