Permalinkhttps://doi.org/10.5802/ahl.156

KeywordsFunctional limit theorems, Poisson process approximation, Kantorovich-Rubinstein distance, Point processes, Stein’s method, Glauber dynamics, Palm coupling, Stabilizing statistics, $k$-nearest neighbor balls, Morse critical points, Binomial point processes

Abstract

We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich–Rubinstein distance using the generator approach to Stein’s method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework – one is for Morse critical points of the distance function, and the other is for large $k$-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thäle (2016) and Otto (2020).

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Permalinkhttps://doi.org/10.5802/ahl.155

KeywordsMeasure equivalence, Orbit equivalence, Amenable groups

Abstract

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of ${\mathrm{L}}^p$ measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results.

On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under ${\mathrm{L}}^1$ measure equivalence. This yields explicit “lower bounds” on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of $3$-solvable groups which pairwise do not coarsely embed into one another.

On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of Følner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under ${\mathrm{L}}^1$ orbit equivalence: the asymptotic dimension and finite presentability.

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Permalinkhttps://doi.org/10.5802/ahl.154

Keywordsinvariant measure, limit distribution, rare events, Poisson process

Abstract

We establish abstract limit theorems which provide sufficient conditions for a sequence $\left(A_l\right)$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the $A_l$ to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to $A_l$ before some time ${\tau }_l$ which is of order $o(1/\mu \left(A_l\right))$. In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.

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Permalinkhttps://doi.org/10.5802/ahl.153

Abstract

We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here “random” means that the coefficients of the polynomials defining the complete intersections are sampled uniformly from the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.

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Permalinkhttps://doi.org/10.5802/ahl.152

Keywordsstochastic processes, Riemannian geometry, group actions

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.151

KeywordsScaling limits, Bessel process, stochastic reinforcement, self-similar Markov process

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.150

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.149

KeywordsRandom planar maps, UIPT, spatial Markov property, pattern occurences

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.148

KeywordsInstanton Floer homology, L-spaces, incompressible tori

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.147

KeywordsMagnetic systems, KAM tori, periodic orbits, trapping regions

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.146

Keywordsinference problem, Hamilton–Jacobi equation, tensor

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.145

Keywordsfractional coloring, Borel sets, Borel combinatorics, Schreier graph, group action, symbolic dynamics

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.144

KeywordsGlobal function fields, Hasse principle, Weak and Strong approximation, Brauer–Manin obstruction, algebraic groups, homogeneous spaces

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.143

KeywordsPlanar maps with three boundaries, generating functions, bijective enumeration, slice decomposition

Abstract

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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Permalinkhttps://doi.org/10.5802/ahl.142

Keywordsespaces de réseaux, points rationnels, groupes orthogonaux, théorème du sous-espace

Abstract

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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