Papers

We study the graph causal optimal transport problem, a generalisation of the classical optimal transport problem in which the allowed couplings satisfy causal restrictions prescribed by a directed graph. We characterise fully the directed acyclic graphs for which the associated graph causal Wasserstein discrepancy is a metric and show that the induced topology agrees with other natural adapted topologies. We characterise the gluing properties of graph causal couplings, prove denseness of Monge couplings, and obtain a dynamic programming principle which allows us to deduce when the graph causal Wasserstein and the adapted Wasserstein distances are equal. Our results link fundamental properties of graph causal optimal transport to structural properties of its underlying graph. Complementing Cheridito and Eckstein (2025), who first introduced such distances and established Lipschitz continuity for the average treatment effect in structural causal models, we obtain Lipschitz continuity of the value function in stochastic team problems.

We construct the multiplicative chaos measures emerging from the Dirichlet series of the twisted Steinhaus multiplicative function. Our work has three main features. First, our argument treats the full subcritical and critical phases simultaneously. Second, we show that the same multiplicative chaos measure arises from any approximation to the Dirichlet series in which the prime cutoff is sent to infinity and the critical line is approached from the right at arbitrary speeds. Such results are known as "universality" results, and we show this via a novel and quick partial summation trick. Third, we show that the multiplicative chaos measure is mutually absolutely continuous with a GMC measure in the same phase, coupled on the same probability space, almost surely. Our work unifies and extends results from a recent breakthrough trilogy of papers by Gorodetsky and Wong.

Everybody learns in school that an out-of-equilibrium system coupled to a heat bath at a fixed temperature evolves to thermodynamic equilibrium as time goes on, and the free energy will only decrease on its way there. At least since Holley's 1971 work, mathematicians know this too, in the modest context of classical Ising Glauber dynamics. But does the free energy also asymptotically decrease to the free energy of an equilibrium state? A typical school kid would say "yes, of course", but since the free energy is only lower semicontinuous, this question is less straightforward than it initially seems. To the best of our knowledge, apart from the uniqueness regime, where one can make use of classical functional inequalities, this question has not previously been resolved rigorously. We prove the asymptotically complete dissipation of the free energy for the classical Ising Glauber dynamics by introducing a coarse modified log-Sobolev inequality, which holds at every positive temperature, in particular in the phase-coexistence regime.

We derive complete local asymptotic expansions near the mode for the negative binomial and hypergeometric laws, complementing the binomial expansion obtained in the companion papers. The coefficients are given in closed Bernoulli-polynomial form and retain the exact lattice displacement from the mean. For each law an entropy normalisation replaces the Bernoulli polynomials by reciprocal Bernoulli polynomials and makes the natural reflection symmetry visible. The first-order coefficient also recovers the exact mode rule: in the Katz cases the vertex gives the classical threshold exactly, while in the hypergeometric case its explicit discrepancy is always too small to change the selected integer mode.

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.

Let $K\subset\mathbb{R}^n$ be an isotropic convex body. We prove that the hit-and-run walk, started from any $M$-warm distribution, reaches total-variation distance $\varepsilon$ from the uniform distribution on $K$ in $O\!\left(n^2\psi_n^{-2}\log^3(M/\varepsilon)\right)$ steps, where $\psi_n^{-1}$ 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 $n^2\psi_n^{-2}$ dependence for hit-and-run, but with polynomial dependence on $M/\varepsilon$. 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.

A stationary Poisson hyperplane process in $\mathbb{R}^d$ 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.

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.

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.

The construction and the stability of bridges between mixture probability measures are investigated. Given a collection of Markov transitions bridging the components of two mixtures, together with a coupling of the mixture labels, we construct a bridge between the mixtures on an extended state space, and we show that this bridge inherits the entropic optimality of the component bridges. We then turn to Gaussian mixtures bridged by Gaussian Schrödinger bridges, whose parameters are generally unknown and must be estimated. We prove a $2$-Wasserstein continuity theorem between the exact bridge and its plug-in approximation. The analysis rests on a representation of the Riccati fixed point map in inverse coordinates; this map is globally $1$-Lipschitz and requires no matrix inversion. These results are applied to three estimation schemes, with $d$ the dimension and $N$ the number of samples. For Gaussian mixtures estimated by the Expectation-Maximization algorithm, the squared $2$-Wasserstein error is of order $(d\log N/N)^{1/2}+d^2\log N/N$ with probability at least $1-10N^{-1}$. For single Gaussian marginals estimated by their sample moments, it is of order $d^2/N$ in expectation. For the regularized empirical Monge map based on an $\epsilon$-inflation of the sample covariance, the mean squared error is of order $d^2\{(1+\epsilon^{-1})/N+\epsilon^2\}$; as soon as $N$ is a sufficiently large multiple of $d$, it reduces to $d^2/N+d\,\epsilon^2$, up to an exponentially small term. These estimates are illustrated numerically.

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent~$\frac{2}{d+2}$ obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\nu$ is a discrete measure and the optimal coupling between $\mu$ and $\nu$ transports each point to a nearest atom of $\nu$, then $W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)$ 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.

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.

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).

In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(\Omega,\mathcal{F})$ and $H$ is an affine set of signed measures on $(\Omega,\mathcal{F})$ 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 $\mathcal{M}$ are the only ones realizing the blow-up of a certain divergence map for any suitable $\phi$-divergence. Finally, considering a different class of $\phi$-divergences, we recover a characterization of the extreme points of $\mathcal{M}$ as strict local maximizers of $\phi$-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.

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 $C^{\theta}_t C^{\infty}_x$ for all $\theta<1/2$ 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 $L^p(L^q)$-regularity, and entropy-entropy dissipation estimates.

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 $\alpha$-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.

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 $\mathbb{R}^d$, 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.

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.

We study Toeplitz determinants $\det T_n(e^f)$ for $f$ whose Fourier coefficients satisfy $f_k=O(|k|^{-1})$. This regime extends beyond $H^{1/2}$ 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 \[ \sum_{k=1}^{\infty}\min(k,n)f_kf_{-k} \] from the higher-order terms in the expansion of $\log\det T_n(e^{tf})$. 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.

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 \cite{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 \cite{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.

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 $L^1$-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.

We consider a branching random walk on \(\Z^d\) in a random environment given by Bernoulli site percolation with parameter \(p\in (0,1)\). In this model, each particle located at an open site reproduces according to a law \(\mu_\circ\), whereas a particle at a closed site reproduces according to another law \(\mu_\bullet\). 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 \((\mu_\circ, \mu_\bullet)\) and establish a Yaglom theorem when both offspring distributions are critical.

We consider the weight-dependent random connection model on a Poisson point process of intensity $\lambda$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\le\beta$. Points at distance $d$ are joined with probability $\min(1,\beta/d)^2$, 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 $\lambda\beta<1$ almost surely all connected components are finite, while for $\lambda\beta\ge31$ an infinite component exists, so at intensity one the critical value satisfies $\beta_c\in[1,31]$; a numerical study included as an appendix places it near $2$. By kernel and profile comparisons the supercritical bound extends to the age-dependent random connection model on the line, which with indicator profile has a non-degenerate phase transition at every value of its parameter, closing a case of the one-dimensional phase diagram left open in earlier work. The lower bound is proved by disconnecting nested pairs of long edges ("rainbows") with cut-point certificates, an argument developed first in a discrete skeleton of the model with the vertices pinned to $\mathbb{Z}$. The skeleton is of independent interest: it has no supercritical phase at all, jumping from total fragmentation to trivial connectivity even though almost surely infinitely many edges cross every fixed site. The supercritical argument is a Peierls argument on the binary tiling of the hyperbolic half-plane.

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.

We introduce a finite stochastic spiking-neuron network with Elephant-type memory, in which past firing activity modifies future excitability through a reinforcement-dependent threshold. For a bounded hard-threshold firing rate, we prove non-explosion of the finite system and obtain conditional exponential contraction in (1)-Wasserstein distance on a truncated potential space. We then formulate the corresponding replica mean-field dynamics and establish global existence, uniqueness in law, and non-explosion of the nonlinear process, together with a characterization of its invariant measures. Numerical experiments show that Elephant memory produces a (p)-dependent decline in firing activity, alters extinction behaviour, and yields finite-network dynamics closely matched by the replica mean-field approximation.

In this paper, we derive quenched scaling limits for linear functionals and the empirical spin field of Ising models on inhomogeneous random graphs generated by a graphon (encompassing both dense and sparse graphs), in the high-temperature regime. We first prove a joint central limit theorem (CLT) for finite collections of linear statistics of the spin configurations, where the limiting covariance is characterized by the resolvent of the associated graphon integral operator. Building on this result, we establish functional CLTs for the average magnetization and for the spin field indexed by suitable classes of regular test functions. We further prove convergence of the full empirical spin field, viewed as a random generalized function in negative Sobolev spaces. These scaling limits provide applications to both Bayesian neural networks and causal inference. Specifically, for the former, we derive infinite-width Gaussian-process limits for two-layer Bayesian neural networks with Ising-dependent output-layer signs, while for the latter, we establish the asymptotic normality of Hájek estimators for average treatment effects under network interference.

Domination of gd-sets, the relation recording which function family approximates which, is closed under box convergence. More generally, approximate 1-Lipschitz maps on asymptotically full-measure subsets, whose restricted measures converge to the target measure after pushforward, place the target representation in every subsequential weak limit of the source pyramids. For weakly convergent pyramids, bounded joint distributions of finite ordered tuples of observables recover both observable diameter, after rightward perturbation of its mass parameter, and the largest common forward gap among several positive-mass sets, after common leftward perturbation of their mass parameters. The lower- and upper-limit formulas agree as the perturbations vanish. Without closure assumptions on a gd-set's function family, we compare observable diameter, the nonnegative part of forward separation, and upper and lower median-tail masses. All observables become uniformly close in measure to suitable constants exactly when both median-tail masses vanish at every positive radius.

We develop a stable convergence theorem for partial sum processes on sample-size dependent stochastic bases. The result allows multidimensional semimartingale limits that have conditionally independent increments and both a continuous and discontinuous martingale part. Motivated by infill asymptotics, it complements classical Gaussian stable limit theorems and supports applications to likelihood based statistical inference.

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.

We study weak limits of multiple Wiener integrals together with the limits of their kernel contractions. After passage to a subsequence, bounded kernel sequences admit a Gaussian Hermite representation whose coordinates carry weights determined by the original chaos orders. In the limits of kernel contractions, only pairings of whole Gaussian coordinates of equal weight contribute. We construct kernel projections with explicit bounds for Gaussian approximation and asymptotic independence. For homogeneous inputs, spectral extraction converges to the decomposable and primitive projections in the limiting kernel spaces. Conversely, prescribed weighted Hermite expansions are realized by smooth compactly supported kernels away from the diagonals, with exact inner products for finite expansions and asymptotically prescribed contractions. We then classify fixed-chaos limits with an absolute exponential moment. For a standard Gaussian vector $X$ and a symmetric matrix $\mathsf A$, a quadratic target $X^{\mathsf T}\mathsf A X-Tr\mathsf A+b^{\mathsf T}X$ is such a limit if and only if $\mathsf A b=0$. We also give a finite cumulant criterion and a joint version. Finally, we identify limits under stationary Ornstein–Uhlenbeck noise: a coordinate of weight $s$ has relaxation rate $s$. Two fourth-chaos examples have identical one-time laws and covariance functions but different asymptotic variances for time averages of their squares. For kernels with nonnegative coordinates, contractions of the weight-one component along simple regular graphs are nonnegative; this excludes an explicit cubic representation that is attained by signed kernels.