We study a self-attractive random walk such that each trajectory of length is penalised by a factor proportional to , where is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately , for some explicit constant . This proves a conjecture of Bolthausen [Bol94] who obtained this result in the case .
We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank . More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances, which also leads to a new description of Patterson–Sullivan distributions.
Typical weighted random simplices , , in a Poisson–Delaunay tessellation in are considered, where the weight is given by the st power of the volume. As special cases this includes the typical () and the usual volume-weighted () Poisson–Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of satisfies a central limit theorem in high dimensions, that is, as . In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight to depend on the dimension as well. A number of special cases are discussed separately. For fixed also mod- convergence and the large deviations behaviour of the logarithmic volume of are investigated.
A Lagrangian subspace of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace of a weak symplectic space which imply that the induced canonical relation from to is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.
We show that the tessellation of a compact, negatively curved surface induced by a long random geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.
For each prime , we study the eigenvalues of a 3-regular graph on roughly vertices constructed from the Markoff surface. We show they asymptotically follow the Kesten–McKay law, which also describes the eigenvalues of a random regular graph. The proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.
We propose a general strategy to derive Lieb–Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb–Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.
We define a partition and a -rotation (-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel–Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition and a -rotation on whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that is a Markov partition for the -rotation on . We prove in both cases that the toral -rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is . The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral -rotations. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.
Motivated by the problem of designing inference-friendly Bayesian nonparametric models in probabilistic programming languages, we introduce a general class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs. More specifically, such a random array is indexed by for some DAG , and its exchangeability structure is governed by the edge set . We prove a representation theorem for such arrays which generalizes the Aldous-Hoover and Austin–Panchenko representation theorems.
This article is concerned with the Schauder estimate for linear kinetic Fokker–Planck equations with Höder continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker–Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.
Gouëzel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.
We argue that Hamilton–Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton–Jacobi equation with asymptotically vanishing viscosity parameter and error term.
In the main result of this paper we prove that a codimension one foliation of , which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.
In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than , where is the Coxeter number of . This provides a replacement for Lusztig’s conjecture, valid under a reasonable assumption on the characteristic.
We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways:
- We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi–Yau 3-folds with codimension 2 discriminant locus.
- We find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.
Cet article donne la description des nœuds des courbes stables qui sont limites de points de Weierstraß. Cela résout le problème posé par Esteves de caractériser les courbes stables dont aucun nœud n’est limite de points de Weierstraß. De plus, nous étendons ce résultat au cas des points de -Weierstraß pour tout . Enfin, nous donnons la description des lacunes de points de -Weierstraß réalisées sur des surfaces de Riemann de genre . Les preuves reposent sur la compactification des strates de différentielles introduites par Bainbridge–Chen–Gendron–Grushevsky–Möller.
We show that the algebraic automorphism group of the character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminations.
Inspirés par l’immeuble de Bruhat–Tits du groupe , pour un corps valué, nous construisons un espace métrique complet sur lequel agit le groupe des automorphismes modérés de l’espace affine. Les points de sont certaines valuations monomiales, et admet une structure naturelle de CW-complexe euclidien de dimension . Quand , et pour de caractéristique zéro, nous prouvons que est localement et simplement connexe, et par conséquent est un espace . En application nous obtenons la linéarisabilité des sous-groupes finis de .
Fix an odd integer . Let be a uniform -angulation with vertices, endowed with the uniform probability measure on its vertices. We prove that there exists such that, after rescaling distances by , converges in distribution for the Gromov–Hausdorff–Prokhorov topology towards the Brownian map. To prove the preceding fact, we introduce a bootstrapping principle for distributional convergence of random labelled plane trees. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton–Watson trees, with only a weak assumption on the centering of label displacements.
We say that a finitely generated group has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the -metric.
We prove that residually finite hyperbolic groups and mapping class groups have (QT).
Let be a convex body in . We consider the Voronoi tessellation generated by a homogeneous Poisson point process of intensity conditional on the existence of a cell which contains . When , this cell converges from above to and we provide the precise asymptotics of the expectation of its defect area, defect perimeter and number of vertices. As in Rényi and Sulanke’s seminal papers on random convex hulls, the regularity of has crucial importance and we deal with both the smooth and polygonal cases. Techniques are based on accurate estimates of the area of the Voronoi flower and of the support function of as well as on an Efron-type relation. Finally, we show the existence of limiting variances in the smooth case for the defect area and the number of vertices as well as analogous expectation asymptotics for the so-called Crofton cell.
We study the -structural stability conjecture from Mañé’s viewpoint for geodesics flows of compact manifolds without conjugate points. The structural stability conjecture is an open problem in the category of geodesic flows because the closing lemma is not known in this context. Without the closing lemma, we combine the geometry of manifolds without conjugate points and a recent version of Franks’ Lemma from Mañé’s viewpoint to prove the conjecture for compact surfaces, for compact three dimensional manifolds with quasi-convex universal coverings where geodesic rays diverge, and for -dimensional, generalized rank one manifolds.
We classify pairs consisting of a complex K3 surface and a finite group such that the subgroup consisting of symplectic automorphisms is among the maximal symplectic ones as classified by Mukai.
For any integer , the -fold cyclic branched cover of an alternating prime knot in the -sphere determines , meaning that if is a knot in the -sphere that is not equivalent to then its -fold cyclic branched cover cannot be homeomorphic to .
In this paper, we provide new concentration inequalities for suprema of (possibly) non-centered and unbounded empirical processes associated with independent and identically distributed random variables. In particular, we establish Fuk–Nagaev type inequalities with the optimal constant in the moderate deviation bandwidth. The proof builds on martingale methods and comparison inequalities, allowing to bound generalized quantiles as the so-called Conditional Value-at-Risk. Importantly, we also extent the left concentration inequalities of Klein (2002) to classes of unbounded functions.
We consider a bounded open Stein subset of a complex Stein manifold of dimension . We prove that if is a current on of bidegree , -closed on , we can find a current on of bidegree which is a solution of the equation in . In other words, we prove that the Dolbeault complex of temperate currents on (i.e. currents on which extend to currents on ) is concentrated in degree . Moreover if is a current on of order , then we can find a solution which is a current on of order .
We establish small-time asymptotic expansions for heat kernels of hypoelliptic Hörmander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by Métivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. Incidentally, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac’s principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.
For general self-similar measures associated with contracting on average affine IFS on the real line, we study the convergence to zero of the Fourier transform at infinity (or Rajchman property) and the extension of results of Salem [Sal44] and Erdös [Erd39] on Bernoulli convolutions. Revisiting in a first step a recent work of Li–Sahlsten [LS19], we show that the parameters where the Rajchman property may not hold are very special and in close connection with Pisot numbers. In these particular cases, the Rajchman character appears to be equivalent to absolute continuity and, when the IFS consists of strict contractions, we show that it is generically not true. We finally provide rather surprising numerical simulations and an application to sets of multiplicity for trigonometric series.
The domino-shuffling algorithm [EKLP92a, EKLP92b, Pro03] can be seen as a stochastic process describing the irreversible growth of a -dimensional discrete interface [CT19, Zha18]. Its stationary speed of growth depends on the average interface slope , as well as on the edge weights , that are assumed to be periodic in space. We show that this growth model belongs to the Anisotropic KPZ class [Ton18, Wol91]: one has and the height fluctuations grow at most logarithmically in time. Moreover, we prove that is discontinuous at each of the (finitely many) smooth (or “gaseous”) slopes ; at these slopes, fluctuations do not diverge as time grows. For a special case of spatially periodic weights, analogous results have been recently proven [CT19] via an explicit computation of . In the general case, such a computation is out of reach; instead, our proof goes through a relation between the speed of growth and the limit shape of domino tilings of the Aztec diamond.
We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.
Let be a polynomial of total degree defined over a perfect field of characteristic zero or greater than . Assuming separable with respect to , we provide an algorithm that computes all singular parts of Puiseux series of above in an expected operations in , where is the valuation of the resultant of and its partial derivative with respect to . To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of in up to an arbitrary precision with arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by with arithmetic operations and, if , with bit operations using probabilistic algorithms, where is the logarithmic height of .
This work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology.
We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3-manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields.
We determine the log-Sobolev constant of the multi-urn Bernoulli–Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant. Our result extends a classical estimate of Lee and Yau (1998) and confirms a conjecture of Filmus, O’Donnell and Wu (2018). Among other applications, we completely quantify the small-set expansion phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.
We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.
We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general.
Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise.
The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Lévy processes with finite Laplace transform.
We show that the Fargues–Fontaine curve associated to an algebraically closed field of characteristic is geometrically simply connected; that is, its base extension from to any complete algebraically closed overfield admits no nontrivial connected finite étale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld’s lemma on the fundamental group of a product of schemes in characteristic .
We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, -Betti numbers, -invariant and some higher-dimensional variants. We also propose various convergence results about -Betti numbers and rank gradient for sequences of actions, groupoids or equivalence relations under weak finiteness assumptions. In particular we connect the combinatorial cost with the cost of the ultralimit equivalence relations. Finally a relative version of Stuck–Zimmer property is also considered.
The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are interested in the relationship between the perimeter (resp. the total curvature, related to the Euler characteristic by Gauss–Bonnet Theorem) of the excursion sets of a function and the ones of its discretization. Our approach is a weak framework in which we consider the functions that map the level of the excursion set to the perimeter (resp. the total curvature) of the excursion set. We will be also interested in a stochastic framework in which the sets are the excursion sets of 2D random fields. We show in particular that, under some stationarity and isotropy conditions on the random field, in expectation, the perimeter is always biased (with a factor), whereas the total curvature is not. We illustrate all our results on different examples of random fields.
We establish variants of the Lefschetz section theorem for the integral tropical homology groups of tropical hypersurfaces of tropical toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in are torsion free. We prove a relationship between the coefficients of the genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge–Deligne numbers of their complex counterparts.
We show that some types for supercuspidal representations of tamely ramified -adic groups that appear in Jiu-Kang Yu’s work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani’s geometrization of the Weil representation and Lusztig’s character sheaves.