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Jérémie Brieussel

Publications and source records attributed to Jérémie Brieussel.

13 recordsLinked to original sources
arXiv2026

Concentration inequalities for maximal displacement of random walks on groups of polynomial growth

We prove Gaussian concentration inequalities for maximal displacement of compactly supported random walks on a compactly generated locally compact group with polynomial growth. Concentration inequalities with different exponents hold for non-centred random walks as well, after correction by the drift. When the support of the measure generates a virtually nilpotent group, we provide an effective version of this result. These more refined estimates rely on the existence of a ``quantitative splitting'' of a virtually simply connected nilpotent group, a result which may be of independent interest. As applications, we deduce that the same concentration inequalities hold for centred random walks on the following classes of groups: amenable connected Lie groups (including non-unimodular ones), polycyclic and more generally finitely generated solvable groups with finite Prüfer rank. This shows in particular that centred random walk are diffusive on such groups. For polycyclic groups, this strengthens and completes partial results previously obtained by Russ Thompson in 2011.

math.PR↗
arXiv2023

Furstenberg entropy spectra of stationary actions of semisimple Lie groups

We determine Furstenberg entropy spectra of ergodic stationary actions of $SL(d,\mathbb{R})$ and its lattices. The constraints on entropy spectra are derived from a refinement of the Nevo-Zimmer projective factor theorem. The realisation part is achieved by means of building Poisson bundles over stationary random subgroups.

math.DS↗
arXiv2021

Noise sensitivity of random walks on groups

A random walk on a group is noise sensitive if resampling every step independantly with a small probability results in an almost independant output. We precisely define two notions: $\ell^1$-noise sensitivity and entropy noise sensitivity. Groups with one of these properties are necessarily Liouville. Homomorphisms to free abelian groups provide an obstruction to $\ell^1$-noise sensitivity. We also provide examples of $\ell^1$ and entropy noise sensitive random walks. Noise sensitivity raises many open questions which are described at the end of the paper.

math.PR↗
arXiv2018

Numerical upper bounds on growth of automata groups

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automata groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automata groups and permitted us to discover several new examples of automata groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

math.GR↗
arXiv2017

Random walks on the discrete affine group

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.

math.GR↗
arXiv2017

Shalom's property $H_{\mathrm{FD}}$ and extensions by $\mathbb{Z}$ of locally finite groups

We show that every finitely generated extension by $\mathbb{Z}$ of a locally normally finite group has Shalom's property $H_{\mathrm{FD}}$. This is no longer true without the normality assumption. This permits to answer some questions of Shalom, Erschler-Ozawa and Kozma. We also obtain a Neumann-Neumann embedding result that any countable locally finite group embedds into a two generated amenable group with property $H_{\mathrm{FD}}$.

math.GR↗
arXiv2016

Speed of random walks, isoperimetry and compression of finitely generated groups

We give a solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups of exponential volume growth. For smaller classes, we give solutions among solvable groups. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the $L_p$-compression exponent of a group and its wreath product with the cyclic group for $p$ in $[1,2]$.

math.GR↗
arXiv2016

Connectedness of spheres in Cayley graphs

We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order $n^2$ to the sphere of radius $n$. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs.

math.GR↗
arXiv2015

Discrete random walks on the group Sol

The harmonic measure $ν$ on the boundary of the group $Sol$ associated to a discrete random walk of law $μ$ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate $μ$ with singular harmonic measure $ν$. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed but singular harmonic measures. The two later results are obtained by comparison with Bernoulli convolutions, using results of Erdös and Solomyak.

math.PR↗
arXiv2013

Behaviors of entropy on finitely generated groups

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq α\leqβ\leq1$, there is a group $Γ$ with measure $μ$ equidistributed on a finite generating set such that \[\liminf\frac{\log H_{Γ,μ}(n)}{\log n}=α,\qquad \limsup \frac{\log H_{Γ,μ}(n)}{\log n}=β.\] The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_n$ on such groups are also evaluated, providing an example of group with return probability satisfying \[\liminf\frac{{\log}|{\log P}(Y_n=_Γ1)|}{\log n}=\frac{1}{3},\qquad \limsup\frac{{\log}|{\log P}(Y_n=_Γ1)|}{\log n}=1\] and drift satisfying \[\liminf\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=\frac{1}{2},\qquad \limsup\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=1.\]

math.GR↗
arXiv2013

Growth behaviors in the range $e^{r^α}$

For every $α\leq β$ in a left neighborhood $[α_0,1]$ of 1, a group $G(α,β)$ is constructed, the growth function of which satisfies $\limsup \frac{\log \log b_{G(α,β)}(r)}{\log r}=α$ and $\liminf \frac{\log \log b_{G(α,β)}(r)}{\log r}=β$. When $α=β$, this provides an explicit uncountable collection of groups with growth functions strictly comparable. On the other hand, oscillation in the case $α< β$ explains the existence of groups with non comparable growth functions. Some period exponents associated to the frequency of oscillation provide new group invariants.

math.GR↗