arXiv subjects

Thomas Budzinski

Publications and source records attributed to Thomas Budzinski.

At least 19 recordsLinked to original sources

arXiv2026

Large deviation principles for pattern-avoiding permutations, and limit shapes for constrained Mallows permutations

We study Mallows random permutations conditioned to avoid a given pattern $\alpha$ of length~$3$. When the bias parameter is of the form $e^{\beta/n}$, we prove that these permutations converge to a non-trivial explicit deterministic permuton that depends on the pattern $\alpha$ and on the parameter $\beta$. Along the way, we provide parametrizations for $\alpha$-avoiding permutons, and establish a large deviation principle for uniform $\alpha$-avoiding permutations. As a byproduct of the proof, we also obtain asymptotic estimates of two versions of $q$-Catalan numbers in the regime $q=e^{\beta/n}$.

math.PR↗

arXiv2025

The longest increasing subsequence of Brownian separable permutons

We establish a scaling limit result for the length $\operatorname{LIS}(\sigma_n)$ of the longest increasing subsequence of a permutation $\sigma_n$ of size $n$ sampled from the Brownian separable permuton $\boldsymbol{\mu}_p$ of parameter $p\in(0,1)$, which is the universal limit of pattern-avoiding permutations. Specifically, we prove that \[\frac{\operatorname{LIS}(\sigma_n)}{n^\alpha}\;\underset{n\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}\; X,\] where $\alpha=\alpha(p)$ is the unique solution in the interval $(1/2,1)$ to the equation \[\frac{1}{4^{\frac{1}{2\alpha}}\sqrt{\pi}}\,\frac{\Gamma\big(\tfrac{1}{2}-\tfrac{1}{2\alpha}\big)}{\Gamma\big(1-\tfrac{1}{2\alpha}\big)}=\frac{p}{p-1},\] and $X=X(p)$ is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent $\alpha(p)$ is an increasing continuous function of $p$ with $\alpha(0^+)=1/2$, $\alpha(1^-)=1$ and $\alpha(1/2)\approx0.815226$, which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter $p\in(0,1)$.

math.PR↗

arXiv2024

The critical Karp--Sipser core of Erd\H{o}s--R\'enyi random graphs

The Karp--Sipser algorithm consists in removing recursively the leaves as well their unique neighbours and all isolated vertices of a given graph. The remaining graph obtained when there is no leaf left is called the Karp--Sipser core. When the underlying graph is the classical sparse Erd\H{o}s--R\'enyi random graph $ \mathrm{G}[n, \lambda/n]$, it is known to exhibit a phase transition at $\lambda = \mathrm{e}$. We show that at criticality, the Karp--Sipser core has size of order $n^{3/5}$, which proves a conjecture of Bauer and Golinelli. We provide the asymptotic law of this renormalized size as well as a description of the distribution of the core as a graph. Our approach relies on the differential equation method, and builds up on a previous work on a configuration model with bounded degrees.

math.PR↗

arXiv2024

Local limits of random bipartite maps in high genus: the general case

We prove the local convergence of uniform bipartite maps with prescribed face degrees in the high genus regime. Unlike in the previous work arxiv:2012.05813 on the subject, we do not make any assumption on the tail of the face degrees, except that they remain finite in the limit.

math.PR↗

arXiv2023

Distances and isoperimetric inequalities in random triangulations of high genus

We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by at most an additive constant. Our main tool to prove those results is an isoperimetric inequality of independent interest: any part of the triangulation whose size is large compared to $\log n$ has a perimeter proportional to its volume.

math.PR↗

arXiv2023

Maximum Agreement Subtrees and H\"older homeomorphisms between Brownian trees

We prove that the size of the largest common subtree between two uniform, independent, leaf-labelled random binary trees of size $n$ is typically less than $n^{1/2-\varepsilon}$ for some $\varepsilon>0$. Our proof relies on the coupling between discrete random trees and the Brownian tree and on a recursive decomposition of the Brownian tree due to Aldous. Along the way, we also show that almost surely, there is no $(1-\varepsilon)$-H\"older homeomorphism between two independent copies of the Brownian tree.

math.PR↗

arXiv2022

The critical Karp--Sipser core of random graphs

We study the Karp--Sipser core of a random graph made of a configuration model with vertices of degree $1,2$ and $3$. This core is obtained by recursively removing the leaves as well as their unique neighbors in the graph. We settle a conjecture of Bauer & Golinelli and prove that at criticality, the Karp--Sipser core has size $ \approx \mathrm{Cst} \cdot \vartheta^{-2} \cdot n^{3/5}$ where $\vartheta$ is the hitting time of the curve $t \mapsto \frac{1}{t^{2}}$ by a linear Brownian motion started at $0$. Our proof relies on a detailed multi-scale analysis of the Markov chain associated to Karp-Sipser leaf-removal algorithm close to its extinction time.

math.PR↗

arXiv2022

On Cheeger constants of hyperbolic surfaces

It is a well-known result due to Bollobas that the maximal Cheeger constant of large $d$-regular graphs cannot be close to the Cheeger constant of the $d$-regular tree. We prove analogously that the Cheeger constant of closed hyperbolic surfaces of large genus is bounded from above by $2/\pi \approx 0.63...$ which is strictly less than the Cheeger constant of the hyperbolic plane. The proof uses a random construction based on a Poisson--Voronoi tessellation of the surface with a vanishing intensity.

math.GT↗

arXiv2021

Multi-ended Markovian triangulations and robust convergence to the UIPT

We classify completely the infinite, planar triangulations satisfying a weak spatial Markov property, without assuming one-endedness nor finiteness of vertex degrees. In particular, the Uniform Infinite Planar Triangulation (UIPT) is the only such triangulation with average degree 6. As a consequence, we prove that the convergence of uniform triangulations of the sphere to the UIPT is robust, in the sense that it is preserved under various perturbations of the uniform measure. As another application, we obtain large deviation estimates for the number of occurencies of a pattern in uniform triangulations.

math.PR↗

arXiv2020

Local limits of bipartite maps with prescribed face degrees in high genus

We study the local limits of uniform high genus bipartite maps with prescribed face degrees. We prove the convergence towards a family of infinite maps of the plane, the q-IBPMs, which exhibit both a spatial Markov property and a hyperbolic behaviour. Therefore, we observe a similar local behaviour for a wide class of models of random high genus maps, which can be seen as a result of universality. Our results cover all the regimes where the expected degree of the root face remains finite in the limit. This follows a work by the same authors on high genus triangulations arXiv:1902.00492.

math.PR↗

arXiv2020

Cooperative and Stochastic Multi-Player Multi-Armed Bandit: Optimal Regret With Neither Communication Nor Collisions

We consider the cooperative multi-player version of the stochastic multi-armed bandit problem. We study the regime where the players cannot communicate but have access to shared randomness. In prior work by the first two authors, a strategy for this regime was constructed for two players and three arms, with regret $\tilde{O}(\sqrt{T})$, and with no collisions at all between the players (with very high probability). In this paper we show that these properties (near-optimal regret and no collisions at all) are achievable for any number of players and arms. At a high level, the previous strategy heavily relied on a $2$-dimensional geometric intuition that was difficult to generalize in higher dimensions, while here we take a more combinatorial route to build the new strategy.

cs.LG↗

arXiv2020

Coordination without communication: optimal regret in two players multi-armed bandits

We consider two agents playing simultaneously the same stochastic three-armed bandit problem. The two agents are cooperating but they cannot communicate. We propose a strategy with no collisions at all between the players (with very high probability), and with near-optimal regret $O(\sqrt{T \log(T)})$. We also argue that the extra logarithmic term $\sqrt{\log(T)}$ should be necessary by proving a lower bound for a full information variant of the problem.

cs.GT↗

arXiv2019

Recurrence of the Uniform Infinite Half-Plane Map via duality of resistances

We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance between the root and the boundary of the hull of radius $r$ is at least of order $\log r$. This resistance bound is expected to be sharp, and is better than those following from previous proofs of recurrence for non bounded-degree planar maps models. Our main tools are the self-duality of uniform planar maps, a classical lemma about duality of resistances and some peeling estimates. The proof shares some ideas with Russo--Seymour--Welsh theory in percolation.

math.PR↗

arXiv2019

The diameter of random Belyi surfaces

We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $n/2$. We show that the diameter of those random surfaces is asymptotic to $2 \log n$ in probability as $n \to \infty$.

math.GT↗

arXiv2019

On the minimal diameter of closed hyperbolic surfaces

We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $\log g$ as $g \to \infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.

math.GT↗

arXiv2019

Infinite geodesics in hyperbolic random triangulations

We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_λ$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton--Watson tree with geometric offspring distribution. The tree of infinite geodesics in $\mathbb{T}_λ$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limits arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper, we obtain new hyperbolicity properties of $\mathbb{T}_λ$: it satisfies a weaker form of Gromov-hyperbolicity and admits bi-infinite geodesics.

math.PR↗

arXiv2019

Universality for random surfaces in unconstrained genus

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov and Pittel (arXiv:1503.01816) have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson--Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov and Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

math.PR↗

arXiv2019

Local limits of uniform triangulations in high genus

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in arXiv:1401.3297. The proof relies on a combinatorial argument and the Goulden--Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane $T$ is weakly Markovian in the sense that the probability to observe a finite triangulation $t$ around the root only depends on the perimeter and volume of $t$, then $T$ is a mixture of PSHT.

math.PR↗