67,310 papers in this slice of arXiv.
Iosif Pinelis
It is shown that, if an event A has the same conditional probability in each trial in an infinite sequence of Bernoulli trials, then A is independent of each trial. More general results are actually established.
Ruizhe Zhang
Let K⊂Rn be an isotropic convex body. We prove that the hit-and-run walk, started from any M-warm distribution, reaches total-variation distance ε from the uniform distribution on K in O(n2ψn−2log3(M/ε)) steps, where ψn−1 is the Kannan-Lovász-Simonovits (KLS) constant. Up to logarithmic factors, this matches the best-known warm-start mixing time for the ball walk. Chen and Eldan [Discrete Comput. Geom. 2026] obtained the same n2ψn−2 dependence for hit-and-run, but with polynomial dependence on M/ε. Our result improves that polynomial dependence to a polylogarithmic one, fully resolving their open question about warm-start mixing of hit-and-run in isotropic convex bodies.
Gilles Bonnet, Eliza O'Reilly, Bharath Roy Choudhury
A stationary Poisson hyperplane process in Rd is characterized by an intensity parameter and an even probability measure on the unit sphere called the directional distribution. In this work, we investigate the stability of the zero cell, i.e., the random convex polytope of the induced hyperplane tessellation containing the origin, under perturbations of the directional distribution. Our results provide quantitative bounds establishing local Hölder continuity of the distribution of the zero cell with respect to Wasserstein metrics on the space of random convex bodies and probability distributions on the unit sphere. As an application, we establish stability bounds for density estimators constructed from Poisson hyperplane tessellations. In particular, we bound the expected total variation distance between the zero-cell-based estimated probability measures in terms of the Wasserstein distance between their directional distributions.
Agus L. Soenjaya
We study the stochastic resistive Hall--magnetohydrodynamic (Hall--MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall--MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.
Harprit Singh
We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions. We apply this framework to parabolic Anderson-type equations associated with positive Rockland operators and with singular initial data. For space-time noise, our results cover the Young regime. For purely spatial noise, we also establish well-posedness in the first singular regime using a variant of the Cole-Hopf transform for Rockland operators. Both applications rely on a mild sewing lemma that accommodates time-dependent Banach spaces and increments with a singularity at the initial time.
Pierre Del Moral, Ajay Jasra, Ke Zhao
In this article we consider bridging between two mixture probability measures. In particular, given access to a Markov kernel between two component distributions, we provide a general mechanism to generate samples from one mixture to the other. Associated to a given reference and extended state space, we prove entropic optimality of this approach. In order to use this idea one needs to know the underlying mixtures and the Markov kernel, which is seldom available, and so we consider the case of Gaussian mixtures and Schrödinger Bridges. We prove a general 2−Wasserstein continuity bound between the exact bridge and one that is approximated, based on ε−covariance inflation, and these rely on a novel continuity analysis of perturbed Riccati maps. We apply our results in the context of bridging mixtures of Gaussians, single Gaussians and empirical estimators of the Gaussian parameters and the Monge map. For mixtures of Gaussians, when the parameters are estimated using the Expectation-Maxmization algorithm, the upper-bound on the 2−Wasserstein distance between the true and approximated bridges is, under assumptions and with probability at least 1−10N−1, O([(NdlogN)1/2{1+(NdlogN)1/2(ε−2+1)}+ε2]) and for the other two cases, in expectation, O(d{1+N1+ε−2+ε2}), where d,N∈N is the dimension of the Gaussian and the number of empirical samples respectively. We also investigate our bounds numerically.
Jonathan Niles-Weed, Jacob Shkrob
We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent d+22 obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every d≥2, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if ν is a discrete measure and the optimal coupling between μ and ν transports each point to a nearest atom of ν, then Wp(μ,ν)≤CdKSWp,1(μ,ν) for a universal constant C, where the complexity parameter K is always at most the number of atoms N and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on k-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on K in this bound cannot be improved, up to polylogarithmic factors.
Aser Cortines, Oren Louidor, Heng Ma +1
We consider a continuous-time random walk on the discrete two dimensional box, driven by the discrete Gaussian Free Field (DGFF) acting as potential: When at a vertex, the walk waits an exponentially distributed time with mean given by the exponential of the field times an inverse temperature parameter and then jumps to one of its neighbors uniformly at random. We prove that when the temperature is below the critical value the walk exhibits ``aging'' at a range of pre-equilibrium time scales: Observed at any such time and then again after an additional time of the same order, there is a positive probability that the walk is found within finite distance from where it was before, with this probability given asymptotically by the Generalized Arcsine Law with a temperature-dependent parameter. We show that this is a consequence of an intricate trapping mechanism which localizes the walk for periods of time which increase with the age of the system, and describe the complex structure of the underlying trapping landscape, which is intimately related to the geometry of the near-extreme level-sets of the DGFF. Altogether, this work demonstrates for the first time an Arcsine-Law aging in the case of a spin-glass-type system with a logarithmically correlated potential, throughout its glassy phase, as predicted in the physic literature.
Quentin Berger, Nicola Turchi, Nikos Zygouras
We estimate the fractional moments of the normalized mass assigned by the Critical 2D Stochastic Heat Flow to small balls. Our results also cover the discrete case corresponding to the 2D directed polymer model and provide estimates that are uniform in all parameters. One key takeaway of our results is that the vanishing of the fractional moments is completely governed by the divergence of the second moment. We use a quite robust method, by refining the change of measure argument and introducing a novel coarse-graining procedure, reducing the proof to essentially second moment estimates (in fact, we also provide sharp second moment estimates for directed polymers, of independent interest).
Gerrit Bauch, Max Nendel, Alessandro Sgarabottolo
In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form M=P∩H, where P denotes the set of all probability measures on an arbitrary measurable space (Ω,F) and H is an affine set of signed measures on (Ω,F) with finite variation. We first give a precise measure-theoretic formulation of the heuristic that extreme measures have minimal support. We then connect this with the notion of minimality with respect to absolute continuity, and prove that points that dominate no other element of M are the only ones realizing the blow-up of a certain divergence map for any suitable φ-divergence. Finally, considering a different class of φ-divergences, we recover a characterization of the extreme points of M as strict local maximizers of φ-divergences relative to any suitable dominating measures. We apply our result to recover and complement results from the literature in the context of finite spaces, sets of measures defined by integral constraints, multi-marginal couplings, and dominated sets of probability measures.
Antonio Agresti, Michael Kniely, Bao Quoc Tang
The existence of global classical solutions for reaction-diffusion systems arising from chemical reaction networks remains a major open problem in the deterministic setting, especially for reactions with high polynomial growth. We prove that a suitably chosen, physically motivated transport noise yields unique strong solutions for complex balanced chemical reaction networks that are global in time with arbitrarily high probability. These solutions possess paths in CtθCx∞ for all θ<1/2 and are, in particular, classical in space. Furthermore, we show that a suitable transport noise can enhance the dissipation of spatial fluctuations at an arbitrarily prescribed exponential rate. Our proofs rely on a combination of scaling-limit arguments, maximal Lp(Lq)-regularity, and entropy-entropy dissipation estimates.
Zbyněk Pawlas, Martina Švarc Petráková
In this paper, we study the random Laguerre tessellation, a weighted generalization of the Voronoi tessellation, generated by a general stationary marked point process. We first derive a nearly optimal sufficient condition on the generating marked point process which ensures that the resulting random Laguerre tessellation is well-defined, utilising the concept of tempered configurations to handle potentially unbounded weights. We then investigate how the three mixing properties - ergodicity, mixing and α-mixing - of the generating marked point process are preserved for the corresponding random Laguerre tessellation. Our approach combines standard approximation arguments with the properties of tempered configurations and the measurability of the Laguerre mapping.
Patrick Forré
We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on Rd, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.
Assaf Shapira
Kinetically constrained lattice gases are interacting particle systems with conserved number of particles, having degenerate rates caused by a kinetic constraint. Over the past decade, there has been major progress in the study of their non-conservative counterpart, kinetically constrained spin models: in dimensions 1 and 2, we have a good understanding of the universality classes of these models. However, the conservative systems are much more challenging to analyze, and very few results are available. The purpose of this paper is to describe universality in the one dimensional case. We will characterize the three universality classes, determining whether such a model is always ergodic, never ergodic, or exhibits an ergodicity phase transition.
Nedialko Bradinoff, Maurice Duits
We study Toeplitz determinants detTn(ef) for f whose Fourier coefficients satisfy fk=O(∣k∣−1). This regime extends beyond H1/2 and includes symbols with Fisher-Hartwig singularities. We develop an operator-theoretic approach based on the Baker-Campbell-Hausdorff formula that separates the quadratic term k=1∑∞min(k,n)fkf−k from the higher-order terms in the expansion of logdetTn(etf). We show that this quadratic term accounts for the possible growth with n, while every fixed higher-order coefficient remains bounded. For symbols with bounded positive and negative Fourier parts, our estimates yield two-sided bounds for the determinant after removal of the quadratic contribution. For a broader admissible class, including Fisher-Hartwig-type symbols, we obtain uniform higher-order coefficient bounds and a central limit theorem for the associated CUE linear statistics. We also obtain bounds on mixed exponential moments for CUE-derived random fields beyond the characteristic polynomial.
Zhanrui Dong, Tiefeng Jiang, Tuan Pham +1
We study the laws of large numbers for the largest eigenvalues among all principal minors of Wishart matrices and deformed GOE matrices. We propose a new method based on identifying the deterministic sets to which the random sets formed by suitably normalized principal minors converge in Hausdorff distance, thereby reducing the original extreme-value problems to finite-dimensional convex optimization problems. We demonstrate the effectiveness of this method in regimes not covered by the existing second-moment arguments in cai2021asymptotic,hu2023extreme. For deformed GOE matrices with fixed minor size k, we determine the limit for every diagonal variance a>0 and identify a phase transition at a=2. Above the transition, the limiting constant satisfies an explicit recursion with no close-form expression, and the optimizers exhibit a nested hierarchical structure, thereby resolving the case left open in cai2021asymptotic. For Wishart matrices with general sub-Gaussian entries and fixed k, we characterize the limit through an entropy-constrained deterministic convex set. When the entries are standard Gaussian, we solve the resulting optimization problem explicitly and obtain the exact value of the limiting constant.
Carlos Cardoso-Perelló, Alberto González-Sanz
We propose a robust barycenter for distribution-valued data by incorporating the Huber loss directly into the optimal transport cost. In contrast to metric-space Huber means, which apply the Huber loss to the Wasserstein distance after optimization, our construction acts on individual transport displacements, preserving quadratic behavior locally while limiting the influence of large displacements. The resulting Huber-Wasserstein barycenters form a natural interpolation between Wasserstein means and L1-type Wasserstein medians. We establish the analytical and statistical foundations of this construction. For optimal transport with Huber loss, we prove regularity and uniqueness properties of dual potentials, existence of optimal transport maps, and stability as the Huber parameter varies. For the associated barycenter problem, we prove existence and characterization results, consistency of empirical plug-in estimators, and a finite-sample breakdown point essentially equal to 1/2. In dimension one, we further derive the pointwise influence function and asymptotic distribution, quantify the associated robustness-efficiency trade-off, and show that displacement-wise Huberization can retain first-order information that is lost by distance-based Huberization under localized shape contamination. Numerical experiments on contaminated distribution-valued data demonstrate the robustness of the proposed barycenters and illustrate their interpolation between mean- and median-like behavior.
Xinxin Chen, Chenlin Gu, Zhiqi Zhao
We consider a branching random walk on Zd in a random environment given by Bernoulli site percolation with parameter p∈(0,1). In this model, each particle located at an open site reproduces according to a law μ∘, whereas a particle at a closed site reproduces according to another law μ∙. Each newly born child performs an independent simple random walk jump from the position of its parent. We study the quenched survival probability under various assumptions on (μ∘,μ∙) and establish a Yaglom theorem when both offspring distributions are critical.
Peter Gracar, Benjamin Lees
We consider the weight-dependent random connection model on a Poisson point process of intensity λ on R×(0,1) in which the vertices (x,t) and (y,s) are joined precisely when (t∨s)∣x−y∣≤β. Points at distance d are joined with probability min(1,β/d)2, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for λβ<1 almost surely all connected components are finite, while for λβ≥31 an infinite component exists, so at intensity one the critical value satisfies βc∈[1,31]; a numerical study included as an appendix places it near 2. The lower bound is proved via a discrete skeleton of the model, obtained by pinning the vertices to Z, which is of independent interest: it has no supercritical phase at all, jumping at a degenerate transition from total fragmentation to trivial connectivity, even though almost surely infinitely many edges cross every fixed site.
Enkelejd Hashorva, Svyatoslav Novikov
In this contribution we study max-stable random fields on the rooted tree under shifts to descendant subtrees. Branch-Brown--Resnick stationarity is characterised through homogeneous spectral classes, punctured tail measures, and local spectral tail fields. For lognormal representers, it is equivalent to invariance of the variogram under addition of a common prefix. We give Gaussian, max-autoregressive, regenerative cascade, and free-group cluster constructions, and show that summability on countably branching trees need not satisfy a zero--one law. We also derive the associated branch-invariant extreme-value and Archimax copulas.