In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate.

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We investigate the effects of noise reinforcement on a Bessel process of dimension $d\in (0,2)$, and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

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Let $E$ be a field of characteristic $p$. The group ${\mathbf{Z}}_p^{\times }$ acts on $E\left(\right(X\left)\right)$ by $a·f\left(X\right)=f({(1+X)}^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of ${\cup }_{n\ge 0}E\left(\left(X^{1/p^n}\right)\right)$. We show how to recover $E\left(\right(X\left)\right)$ from the valued $E$-vector space $\tilde{\mathbf{E}}$ endowed with its action of ${\mathbf{Z}}_p^{\times }$. To do this, we introduce the notion of super-Hölder vector in certain $E$-linear representations of ${\mathbf{Z}}_p$. This is a characteristic $p$ analogue of the notion of locally analytic vector in $p$-adic Banach representations of $p$-adic Lie groups.

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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.

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We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton $L$-spaces, by a map which identifies meridians with Seifert longitudes, cannot be an instanton $L$-space. This recovers the recent theorem of Lidman–Pinzón-Caicedo–Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial $\mathit{SU}\left(2\right)$-representation, and consequently Zentner’s earlier result that the fundamental group of every closed, oriented $3$-manifold besides the 3-sphere admits a nontrivial $\mathit{SL}(2,ℂ)$-representation.

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We study non-resonant circles for strong magnetic fields on a closed, connected, oriented surface and show how these can be used to prove the existence of trapping regions and of periodic magnetic geodesics with prescribed low speed. As a corollary, there exist infinitely many periodic magnetic geodesics for every low speed in the following cases: i) the surface is not the two-sphere, ii) the magnetic field vanishes somewhere.

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We consider a general statistical inference model of finite-rank tensor products. For any interaction structure and any order of tensor products, we identify the limit free energy of the model in terms of a variational formula. Our approach consists of showing first that the limit free energy must be the viscosity solution to a certain Hamilton–Jacobi equation.

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Let $\Gamma$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift $\Gamma \curvearrowright 2^{\Gamma }$ (with respect to some finite subset $F\subseteq \Gamma$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $\Gamma$ is the free group, answering a question of Meehan.

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Let $X$ be a homogeneous space of a reductive group with reductive stabilizers, defined over a global field of positive characteristic. Using duality theorems for complexes of tori, we study cohomological obstructions to various arithmetic properties.

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We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for tight pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit.

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Nous étudions l’exposant diophantien d’un point $x$ d’une hypersurface quadratique. Nous montrons notamment un analogue du théorème de Thue–Siegel–Roth, c’est-à-dire une formule pour l’exposant diophantien d’un point algébrique, et un analogue du résultat de Kleinbock et Margulis sur l’extrémalité des sous-variétés non dégénérées de l’espace affine.

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We consider the level-sets of continuous Gaussian fields on ${ℝ}^d$ above a certain level $-\ell \in ℝ$, which defines a percolation model as $\ell$ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than $d$ (in particular, this includes the Bargmann–Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point ${\ell }_c$. More precisely, we show that connection probabilities decay exponentially for $\ell <{\ell }_c$ and percolation occurs in sufficiently thick 2D slabs for $\ell >{\ell }_c$. This extends results recently obtained in dimension $d=2$ to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice $\epsilon {\mathbb{Z}}^d$) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter $\ell$.

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We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time $t$, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978). An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process. Our proofs incorporate new ideas which might be of use in other branching models.

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It is known that for every $\alpha \ge 1$ there is a planar triangulation in which every ball of radius $r$ has size $\Theta \left(r^{\alpha }\right)$. We prove that for $\alpha <2$ every such triangulation is quasi-isometric to a tree. The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs. We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree.

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For $\alpha \in (1,2]$, the $\alpha$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given $\alpha$-dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an $ℝ$-tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning $ℝ$-tree, which is a biased version of the $\alpha$-stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals, and observe that these distributions are themselves related to the distributions of certain configuration models; (2) determine its distribution as a collection of $\alpha$-stable trees glued onto its kernel; and (3) present a line-breaking construction, in the same spirit as Aldous’ line-breaking construction of the Brownian continuum random tree.

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