Permalinkhttps://doi.org/10.5802/ahl.131

KeywordsHeavy atoms, ground state density, relativistic Coulomb system, Scott conjecture

Abstract

We prove the convergence of the density on the scale $Z^{-1}$ to the density of the Bohr atom with infinitely many electrons (strong Scott conjecture) for a model that is known to describe heavy atoms accurately.

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Permalinkhttps://doi.org/10.5802/ahl.130

KeywordsRandom walk; cone; exit time; Green function; harmonic function; Martin boundary; Brownian motion; coupling

Abstract

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

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Permalinkhttps://doi.org/10.5802/ahl.129

KeywordsHypocoercivity, Langevin dynamics, Schur complements, resolvent estimates

Abstract

We propose an approach to obtaining explicit estimates on the resolvent of hypocoercive operators by using Schur complements, rather than from an exponential decay of the evolution semigroup combined with a time integral. We present applications to Langevin-like dynamics and Fokker–Planck equations, as well as the linear Boltzmann equation (which is also the generator of randomized Hybrid Monte Carlo in molecular dynamics). In particular, we make precise the dependence of the resolvent bounds on the parameters of the dynamics and on the dimension. We also highlight the relationship of our method with other hypocoercive approaches.

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Permalinkhttps://doi.org/10.5802/ahl.128

KeywordsHighly transitive actions, infinite permutation groups, confined subgroups and uniformly recurrent subgroups, groups of piecewise linear homeomorphisms, Higman–Thompson groups

Abstract

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k\ge 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points.

This result provides a tool to rule out the existence of highly transitive actions, and to classify highly transitive actions of a given group. We give concrete illustrations of these applications in the realm of groups of dynamical origin. In particular we obtain the first non-trivial classification of highly transitive actions of a finitely generated group.

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Permalinkhttps://doi.org/10.5802/ahl.127

KeywordsHartree equation, nonlinear Schrödinger equation, density matrices, random fields, stability, scattering

Abstract

We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions $2$ and $3$, extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrödinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger equation, and on the use of explicit low frequency cancellations as done by Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators.

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Permalinkhttps://doi.org/10.5802/ahl.126

Keywordsnormalized gradient flow, ground states, stationary states, quantum graphs, nonlinear Schrödinger equation

Abstract

We introduce and implement a method to compute stationary states of nonlinear Schrödinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schrödinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

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Permalinkhttps://doi.org/10.5802/ahl.125

KeywordsRandom maps, random trees, scaling limits

Abstract

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit nontrivial subsequential limits as $n\to \infty$ in the Gromov–Hausdorff–Prokhorov topology. Further, we show that they converge in distribution towards the celebrated Brownian sphere, and more generally a Brownian disk for maps with a boundary, if and only if there is no inner face with a macroscopic degree, or, if the perimeter is too big, the maps degenerate and converge to the Brownian tree. By first sampling the degrees at random with an appropriate distribution, this model recovers that of size-conditioned Boltzmann maps associated with critical weights in the domain of attraction of a stable law with index $\alpha \in [1,2]$. The Brownian tree and disks then appear respectively in the case $\alpha =1$ and $\alpha =2$, whereas in the case $\alpha \in (1,2)$ our results partially recover previous known ones. Our proofs rely on known bijections with labelled plane trees, which are similarly sampled uniformly at random given $n$ outdegrees. Along the way, we obtain some results on the geometry of such trees, such as a convergence to the Brownian tree but only in the weaker sense of subtrees spanned by random vertices, which are of independent interest.

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Permalinkhttps://doi.org/10.5802/ahl.124

KeywordsSchreier decoration, Borel graphs, local-global equivalence, edge colorings

Abstract

Following recent result of L. M. Tóth [arXiv:1906.03137] we show that every $2\Delta$-regular Borel graph $𝒢$ with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate Kőnig’s line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.

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Permalinkhttps://doi.org/10.5802/ahl.123

Keywordsgrowth-fragmentation equation, self-similar fragmentation, measure solutions, long-time behavior, general relative entropy, Harris’s theorem, periodic semigroups

Abstract

We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptotic oscillations and that the convergence to this periodic behavior, in suitable spaces of weighted signed measures, occurs exponentially fast. It can be seen as a spectral gap result between the countable set of dominant eigenvalues and the rest of the spectrum, which is to our knowledge completely new. The two main difficulties in the proof are to define the projection onto the subspace of periodic (rescaled) solutions and to estimate the speed of convergence to this projection. The first one is addressed by using the generalized relative entropy structure of the dual equation, and the second is tackled by applying Harris’s ergodic theorem on sub-problems.

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Permalinkhttps://doi.org/10.5802/ahl.122

KeywordsStochastic regularization, Stochastic Partial Differential Equations, Nonlinear Schrödinger equation, Hardy–Littlewood–Sobolev inequality

Abstract

In this paper, we investigate a stochastic Hardy–Littlewood–Sobolev inequality. Due to the non-homogenous nature of the potential in the inequality, we show that a constant proportional to the length of the interval appears on the right-hand-side. As a direct application, we derive local Strichartz estimates for randomly modulated dispersions and solve the Cauchy problem of the critical nonlinear Schrödinger equation.

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Permalinkhttps://doi.org/10.5802/ahl.121

KeywordsDouble bracket, Hamiltonian algebra, Quasi-Hamiltonian algebra, Non-commutative geometry, Integrable system

Abstract

Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the $H_0$-Poisson structures of Crawley–Boevey. We prove in particular that the double (quasi-) Poisson algebra structure defined by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as an undirected graph, up to isomorphism. We derive from our results a representation theoretic description of action-angle duality for several classical integrable systems.

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Permalinkhttps://doi.org/10.5802/ahl.120

KeywordsLinear Differential Equations, Differential Galois Theory, Algebraic Relations, Iterated Integrals, Hypergeometric Functions

Abstract

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular, if $f$ and $g$ satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to $f$ guaranteeing that $g$ is a polynomial in $f$. We apply this to hypergeometric functions and iterated integrals.

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Permalinkhttps://doi.org/10.5802/ahl.119

Keywordscharacter variety, Borel-de Siebenthal subgroups, free group, homotopy groups, singularities

Abstract

Let $G$ be a connected reductive complex affine algebraic group, and let ${𝔛}_r$ denote the moduli space of $G$-valued representations of a rank $r$ free group. We first characterize the singularities in ${𝔛}_r$, extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino–Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel–de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of ${𝔛}_r$, proving conjectures of Florentino–Lawton–Ramras. Lastly, using the earlier analysis of Borel–de Siebenthal subgroups, we prove a conjecture of Sikora about centralizers of irreducible representations in Lie groups.

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Permalinkhttps://doi.org/10.5802/ahl.118

KeywordsGRR, DRR, ORR, Cayley graph, automorphisms of graphs, generalized dihedral group, generalized dicyclic group, regular automorphism group

Abstract

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs.

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Permalinkhttps://doi.org/10.5802/ahl.117

KeywordsReaction-diffusion, Dispersal evolution, Front acceleration, Linear determinacy, Approximation of geometric optics, Lagrangian dynamics, Explicit rate of expansion

Abstract

We investigate super-linear spreading in a reaction-diffusion model analogous to the Fisher-KPP equation, but in which the population is heterogeneous with respect to the dispersal ability of individuals and the saturation factor is non-local with respect to one variable. It was previously shown that the population expands as $𝒪\left(t^{3/2}\right)$. We identify a constant ${\alpha }^{*}$, and show that, in a weak sense, the front is located at ${\alpha }^{*}{t}^{3/2}$. Surprisingly, ${\alpha }^{*}$ is smaller than the prefactor predicted by the linear problem (that is, without saturation) and analogous problem with local saturation. This hindering phenomenon is the consequence of a subtle interplay between the non-local saturation and the non-trivial dynamics of some particular curves that carry the mass to the front. A careful analysis of these trajectories allows us to characterize the value ${\alpha }^{*}$. The article is complemented with numerical simulations that illustrate some behavior of the model that is beyond our analysis.

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