Papers

We prove that the very-high temperature long-range Ising model on $\mathbf{Z}^d$ is a factor of an i.i.d. process, and provide quantitative information on the coding radius. We do not assume that the model is ferromagnetic, but require the interactions to decay faster than $1/n^{2d}$. While previous constructions were known, the novelty of our approach is that we obtain an explicit polynomial upper bound on the tail of the coding radius. In particular, our construction yields a factor map with finite expected coding volume. This answers a question of a recent preprint by Chazottes, Gallo and Takahashi, see arXiv:2602.14618.

Rapid algorithmic developments have accelerated the sampling of redistricting ensembles (balanced graph partitions), yet evaluating rare events and sampling complex target measures remains a core challenge due to the high-dimensional and combinatorial nature of the phase space. We address a prerequisite for stratified sampling on this space: constructing and diagnosing candidate strata with suitable coverage and overlap. We build a grammar on observed plans by clustering districts into representative “letters” and using them to form plan-level “words.” A partition of unity over these words gives a soft assignment of plans to strata and allows us to estimate stratum masses and an overlap-induced flux matrix. We demonstrate this computational pipeline using real-world congressional redistricting data from Connecticut and examine how strata learned from one target distribution behave under related distributions. The resulting construction provides a foundation for future stratified sampling on spaces of redistricting plans or balanced graph partitions. We do not implement a complete stratified sampler here; evaluating whether the proposed strata improve sampling efficiency or reduce estimator variance is left for future work.

We give an approximately uniform sampler for satisfying assignments of constraint satisfaction problems that satisfy $4\mathrm e p(\Delta+1)^2\le1$, where $p$ is the largest constraint-violation probability under the uniform product distribution, and $\Delta$ is the maximum degree of the dependency graph. The algorithm invokes the recent efficient approximate counting algorithm of Liu, Wang, Yin, Zhang, and Zhou~\cite{CountingLLL} as a subroutine and returns a satisfying assignment sampled within total-variation distance $\varepsilon$ of the uniform distribution in $(n+m/\varepsilon)^{O(k\Delta\log D)}$ time, where $n$ and $m$ are the numbers of variables and constraints, $D$ is the common domain size, and $k$ bounds the constraint arity. We also give an asymmetric product-form condition under which both counting and sampling are efficient. To supply the counts needed in the additional regime, we extend the marginal-expansion proof of Liu et al.

Computing the total variation (TV) distance between succinctly represented high-dimensional distributions is generally intractable. We give an FPRAS for TV distance between mixtures of product distributions and, more generally, for a natural class of structured probabilistic circuits. Our main technique is a novel application of domain reduction: Given a family of feature vectors indexed by assignments, we use Lewis-weight sampling to replace the assignment domain by a polynomial-size weighted subset that simultaneously approximates the sum of absolute values of every linear projection. For mixtures of product distributions, we construct such reduced domains incrementally over the coordinates, obtaining the first FPRAS with running time polynomial in both the dimension and the number of mixture components. We then extend the approach to smooth, structured-decomposable probabilistic circuits with a common structured architecture.

We study the expected number of zeros of the Short-Time Fourier Transform (STFT) with a Gaussian window for signals degraded by complex Gaussian colored noise. We provide an exact formula for this quantity and investigate its asymptotic behavior. From a computational perspective, we propose an algorithmic method to recover a smoothed power spectral density (PSD) directly from the spatial distribution of the transform's zeros. Our results formalize the commonly-held heuristic that detection algorithms based on spectrogram zeros, though designed for white noise, also perform adequately under moderately colored noise.

In this paper, we derive a new lower bound for the Hausdorff dimension of 3D Brownian cut points by proving an explicit upper bound ($<0.9999$) for $\xi_3(1,1)$, the intersection exponent for two independent Brownian motions in 3D. Consequently, the growth exponent of 3D loop-erased random walk is at least $1.0001$.

This paper aims to develop efficient numerical approximations for a class of stochastic differential equations with state-dependent fast switching processes. The direct Euler–Maruyama (EM) scheme fails when the scaling parameter is small. Based on the heterogeneous multiscale method of \cite{e2005analysis}, we propose three algorithms and prove their strong $L^p$-convergence with explicit rates for any $p\geq 2$. In the first algorithm, we combine the averaging principle with an EM scheme for the averaged equation, where the invariant measure of the Markov chain can be explicitly obtained by solving a linear system. However, computing this invariant measure incurs cubic cost as the number of switching states increases. To avoid solving large linear systems, we approximate the invariant measure instead. Thus the second and third algorithms both combine a macroscopic EM scheme for a modified averaged equation with micro-solvers that estimate the averaged drift. More precisely, in the second algorithm, a discrete-time Markov chain is simulated with a micro time step, and the averaged drift is obtained by averaging over finitely many micro transitions. However, both the first and second algorithms only work when the switching process has finite states. Therefore, we introduce a third algorithm that allows for switching processes with countably infinite states, in which exact continuous-time Markov chain is generated via the Gillespie algorithm, and the averaged drift is computed by exact time averaging over a specified interval. Numerical experiments verify the theoretical results and demonstrate the computational advantages of these three algorithms.

In 2009, Yano, Yano and Yor asked whether the $\alpha$-Cauchy distributions are self-decomposable or at least infinitely divisible in their study of hitting times for symmetric stable processes. Recently, the first named author proved that $\alpha$-Cauchy distribution is infinitely divisible if and only if $1 < \alpha \leq 2$. In this paper, we prove that $\alpha$-Cauchy distribution is self-decomposable if and only if $\alpha = 2$. The proof is based on a new criterion for self-decomposability in the symmetric case. As an application, we show that the first hitting time of a nonzero point by a symmetric stable process of index $1<\alpha<2$ starting from zero is not self-decomposable.

We study the asymptotic behavior of a sequence of Galton–Watson processes with inhomogeneous immigration when the limit of the means of the offspring distributions is less than $1$ or equal to $1$. Under growth conditions on the expected values of the immigration distributions and the variances of the offspring distributions, and assuming the weak convergence of properly scaled immigration processes towards a non-negative stochastic process $\mathcal Y$ with càdlàg or continuous sample paths, we establish functional limit theorems for the sequence of Galton–Watson processes with inhomogeneous immigration in question. The limit stochastic processes can be represented as a constant multiple or an integral functional of $\mathcal Y$.

We explain how to derive the Robinson-Schensted-Knuth (RSK) correspondence, a fundamental bijection in algebraic combinatorics, from the Yang-Baxter equation. The Yang-Baxter equation arose in the study of quantum many-body systems and later became a cornerstone of the theory of solvable lattice models, particularly vertex models. In a vertex model, arrows occupy the edges of a grid, and each vertex carries a Boltzmann weight determined by the arrows on the four edges meeting at it. The weight of a configuration is the product of these local weights, and a partition function is the sum of the weights of all configurations with prescribed boundary conditions. For a vertex model whose partition functions are the Schur polynomials, the two sides of each instance of the Yang-Baxter equation admit exactly one weight-preserving matching of their summands. Carried across a grid, this forced matching is the classical RSK correspondence in the form of Fomin's growth diagrams. In natural coordinates the local matching becomes the combinatorial three-dimensional R, a set-theoretic solution of the Zamolodchikov tetrahedron equation. Its periodic closure returns the combinatorial R-matrices of one-row crystals. The forced matching is special to the Schur weights. For the Hall-Littlewood and q-Whittaker deformations and their spin versions, at generic parameter values no deterministic matching works for all boundary data. Reading each instance of the Yang-Baxter equation probabilistically, we replace the matching by a coupling of the two sides - a bijectivization, or probabilistic bijection - and obtain Markov operators that transport probability measures attached to vertex models. Iterated over the grid, these operators produce randomized RSK-type dynamics and interacting particle systems, including q-PushTASEP and the stochastic six-vertex model.

Recently, C. Li posted a preprint that resolves the fractional version of Talagrand's discrete convexity problem. In this note, we use the method of C. Li to answer another problem of Talagrand.

We prove the half-plane one-arm exponent for critical Bernoulli bond percolation on the square lattice. Ikhlef and Ponsaing obtained an exact formula for the probability that the unique infinite percolation hull running along an odd-width strip passes through a prescribed boundary edge. Their interpolation, however, treats a rational function as a polynomial. We clear the full denominator, prove a coordinatewise Laurent-degree bound directly at $q^3=1$, and establish the exact strip formula from the deletion relations. At the homogeneous point, the qKZ expression is the Bernoulli probability of this boundary-passage event. A finite local modification, the RSW theorem, and one-arm quasi-multiplicativity then give, uniformly for $1\le r<R$, ${\bf P}_{1/2}(A_1^+(r,R))\asymp (r/R)^{1/3}$.

Let $d>1$ be fixed and let $A_n$ be an $n\times n$ matrix with independent $\Ber(d/n)$ entries. For every $0<r<\sqrt d$, we prove that, with high probability, a positive proportion of the eigenvalues of $A_n$ have modulus larger than $r$. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to $\sqrt d$. Our proof works with the Brown measure $\mu_d$ of the adjacency operator of the directed Poisson–Galton–Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of $A_n$ to $\mu_d$ in probability. We prove that the outer radius of its support is $\sqrt d$, using a resolvent recursion on Poisson–Galton–Watson trees.

We study statistical variants of the classical largest average submatrix problem. For small submatrices (where the dimension is less than linearly proportional to the original matrix), the problem is typically well understood and believed to exhibit the statistical-computational gap (SCG). However, analytical treatment of both the information-theoretic and algorithmic aspects of the linear regime remains challenging, and no mathematically rigorous results have yet arrived anywhere close to proving or disproving SCG existence in this setting. Focusing on the linear regime, we make strong progress in several key directions: 1) We develop a generic Random Duality Theory (RDT) framework to characterize largest average submatrix values. 2) Using the plain RDT variant, we obtain closed-form upper bounds as explicit functions of dimensionality proportionality parameters. 3) We demonstrate that a lifted RDT variant strictly improves upon the plain RDT within a certain linear range of submatrix dimensions. 4) For small submatrices where the dimensional proportionality constants approach zero, we prove that our results match both the replica results from [34] (obtained via one-step replica symmetry breaking) and the sublinear results from [18,45].

We perform a detailed study of a principal–agent problem in a continuous time version of the celebrated Holmström–Milgrom model (Econometrica 55 (2), 1987) where we add limited liability for the Agent. We develop a probabilistic methodology to prove that the Principal's value function is the unique bounded classical solution to a fully nonlinear and fully degenerate partial differential equation (PDE) with Cauchy-Dirichlet boundary conditions on $[0,T]\times[0,\infty)$. Indeed, we also prove infinite continuous differentiability of the solution in the interior of the domain. The strength of our regularity result is such that we can ensure existence of optimal controls in strong form—a rare occurrence in dynamic contracting—and we obtain fine properties of the optimal control map, including a characterisation via a further nonlinear degenerate PDE.

We establish a Levy interpretation of normalized forward BFKL small-$x$ evolution at leading order and fixed coupling, and exclude a probability law at fixed-order next-to-leading accuracy in the symmetric scheme. After the Marchesini-Onofri conjugation and growth subtraction, one positive step measure in closed form covers all conformal spins, on the cylinder of logarithmic transverse momentum and azimuth. The process is pure jump, and the Pomeron intercept is the relaxation rate of its first azimuthal harmonic. With $s_0=q_1q_2$ and the coupling at that argument, the negative cubic collinear pole of the truncated eigenvalue excludes a probability law at every positive coupling and rapidity. Its leading coefficient is independent of $N_c$ and $n_f$. The pole dominates the leading-order simple pole within $\sqrt{\bar\alpha/2}$ of the edge in the leading collinear approximation. Translation-preserving multiplicative conjugations cannot restore positivity. No non-negative step measure generates that truncated evolution, while a positive radial completion at zero conformal spin matches the computed order, so the obstruction is a property of the fixed-order truncation. The tested symmetric-scheme resummations also fail positivity, as shown in closed form for the pure and matched all-poles forms, except for a degenerate zero process, and by computation at the examined couplings for the full prescription. The improved finite-rapidity Green function fails at the displayed parameters under the contour assumption. A resummed kernel in another rapidity scheme has a non-negative step measure at tested couplings. An unweighted transverse walk describes the leading-order evolution with no approximation, but a probabilistic resummation of the symmetric next-to-leading kernel must establish positivity beyond perturbative matching, and whether one exists is left open.

This paper provides a recursive formula for computing the coefficients in a small-system-noise asymptotic expansion for nonlinear filtering. The expansion, obtained from the Kallianpur–Striebel formula, was justified in the author's previous work. Our main contribution is to reduce the coefficient calculation to a finite-dimensional system extending the Kalman–Bucy filter by applying Fubini's theorem and Wick's formula and differentiating the resulting terms. For each fixed expansion order, the number of variables grows at most polynomially, rather than exponentially, with the system dimension. To justify the construction, we define the required non-adapted integrals as limits of discrete sums and establish a generalized Ito formula. We also extend the expansion from conditional expectations to conditional characteristic functions and provide a numerical illustration of the method.

We give a direct proof of the Burke property for the stationary O'Connell-Yor polymer using Stein's method. Building on a free energy identity developed in joint work with Firas Rassoul-Agha, Xiao Shen, and Guangqu Zheng, we establish a multilevel extension that yields a joint Gaussian-Gamma Stein identity, from which the full Burke property follows. The proof is based on a Gaussian integration-by-parts framework developed in that work. Passing to zero temperature gives the corresponding result for stationary point-to-line Brownian last-passage percolation.

We develop a Dirichlet-form framework for relaxation and steady-state entropy production in preconditioned Langevin stochastic partial differential equations. In infinite dimensions, the usual Fokker–Planck calculations based on ambient Lebesgue densities and the corresponding probability-current densities are generally unavailable. For the detailed-balance class, a quasi-regular symmetric Dirichlet form on the Gibbs state space yields an exact de Bruijn entropy-dissipation formula for regular densities and an integrated inequality for arbitrary finite-entropy initial laws. Self-adjointness gives detailed balance and stationary path reversal, while a coordinate-martingale criterion identifies the associated process with the prescribed SPDE. For the one-dimensional $\Phi^4_1$ and convex Allen–Cahn-type Gibbs dynamics, we combine the established strong well-posedness theory with direct verification of the logarithmic derivatives, form closure, quasi-regularity and form–SPDE correspondence, and obtain relative-entropy decay bounds with exponents $2$ and $2(1-\lambda/m)$, respectively, with $m>\lambda$ in the latter case. Away from detailed balance, bounded skew-adjoint linear forcing preserves a Gaussian invariant law without requiring commutation between the forcing and covariance. We identify the antisymmetric action on cylinder observables and its Cameron–Martin current and prove that the squared current energy equals the steady-state entropy-production rate defined by forward–reverse path-space relative entropy per unit time, as well as the monotone limit of the Galerkin rates. Under commutation, we additionally obtain an explicit transient Onsager decomposition after a mass quench. Exclusion processes and Gaussian rotors provide finite-state and Gaussian benchmarks.

We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_q)}))$, of the form $a_0+a_1\frac{1}{n}+O(\frac{1}{n^2})$. This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which $x_{i,j}$ is complex Gaussian.

We establish three weak averaging principles for distribution-dependent stochastic Navier–Stokes equations with rapidly oscillating coefficients on the torus $\T^2$. First, solution laws are relatively compact on finite time intervals, and every limit point as $\var\to0$ is the path law of a variational martingale solution of the averaged equation. Under a dissipative condition, we can choose bounded complete variational solution laws of the original and averaged equations so that every sequence with $\var\to0$ has a subsequence converging weakly to a bounded complete variational solution law of the averaged equation. At the level of probability laws, the original nonautonomous equation admits a family of weak pullback attractors, while the averaged equation has a weak global attractor. The weak pullback attractors converge upper semicontinuously to the weak global attractor of the averaged equation, uniformly with respect to the coefficient hull.

We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.

We introduce a one-parameter family of notions between double ergodicity and metric ergodicity for measure-class preserving (i.e., nonsingular) actions of countable groups on standard probability spaces, providing infinitely many new invariants distinguishing weakly mixing actions. We call these properties (essentially) $k$-chaining, for $k \in \mathbb{N}$. We apply this framework to study boundary actions of free groups of finite rank $r \ge 1$, where the boundary is equipped with a stationary Markov measure. We prove that in this context, weak mixing is equivalent to $(2r-1)$-chaining, as well as to strict irreducibility of the transition matrix of the Markov measure.

Chatterjee and Sloman proved that a bounded measurable similarity function with sufficiently small average Gromov hyperbolicity admits a tree representation with small mean approximation error. Their argument uses a weighted version of Szemerédi's regularity lemma and does not yield explicit quantitative bounds. Here, we establish a tighter relation between average hyperbolicity and mean tree approximation error. For a similarity function $s:S\times S\to[0,b]$, we prove that $$\operatorname{Tree}(s) \leq (63/e)^{1/3} \sqrt[3]{b^2 \operatorname{Hyp}(s)} \leq 2.8512 \sqrt[3]{b^2 \operatorname{Hyp}(s)}.$$ The proof uses a simple pivoting construction inspired by \KwikCluster. We also discuss the optimal dependence on average hyperbolicity, including a square-root lower bound, and connections with ultrametric fitting.

We present a solution of the limiting five-expert prediction problem with geometric stopping, based on stochastic calculus and partial differential equations. If $\delta$ is the stopping probability, the minimax expected regret from tied initial scores is $45\pi^2/(512\sqrt{2\delta})+o(\delta^{-1/2})$ as $\delta\downarrow0$. The adversarial rank control selecting the best and third-best experts maximizes the limiting Hamiltonian at every state. Following the four-expert construction of Bayraktar, Ekren, and Zhang (2020), we represent the value correction as a discounted boundary-local-time expectation for a degenerate obliquely reflected Brownian motion. The hyperbolic systems governing its boundary traces are derived and solved explicitly. To verify the nonlinear Hamilton–Jacobi–Bellman equation, we combine heat-equation minimum principles in equality directions, ODE minimum principles applied to convex combinations of multiple controls, and comparisons across controls. These arguments reduce the 48 regional control inequalities to a few lower-dimensional boundary problems, with the remaining signs established by explicit monotonicity arguments and by factorization of multivariate integral kernels that are positive polynomials in some of their variables. We prove global $C^2$ regularity across both regional interfaces and changes in rank ordering. We also show that the alternating COMB control attains the limiting Hamiltonian maximum exactly when the two highest scores coincide and the third- and fourth-highest scores coincide.

We consider a soft version of the spherical Sherrington-Kirkpatrick model, where the spherical constraint is replaced by a radial confinement term. We prove that, for a real symmetric disorder matrix under suitable spectral assumptions, the free energy exhibits two distinct fluctuation regimes. In the high-temperature regime, the fluctuations are asymptotically Gaussian. In the low-temperature regime, the fluctuations are governed by the largest eigenvalue and converge to the GOE Tracy-Widom distribution. These results show that the soft radial constraint preserves the fluctuation transition of the classical spherical Sherrington-Kirkpatrick model.

We reformulate average-cost (ergodic) control of a Markov jump process as a finite-horizon stochastic optimal transport (SOT) problem that jointly optimizes over the controlled evolution and the marginal law from which it starts and returns to (i.e., a self-transport). The constraint relating the marginal flow to the controlled generator is the basic adjoint relationship (BAR), so we call the resulting problem BAR-SOT. For any time horizon T>0, its optimal value is a constant scaling of the long-run average-cost rate. We give three equivalent formulations (through controlled processes, a Fokker–Planck constraint, and relaxed marginal measures) and show that the optimal dual is a stationary potential plus a term linear in time, with slope equal to the rate. A relative-entropy penalty on the control yields a cost-tilted Schrodinger bridge problem, computable by a Sinkhorn-type iteration when every transition rate is controlled, whose value converges to the unregularized optimum as the penalty vanishes. We develop the theory for finite Markov decision processes and then for general controlled Markov jump processes. A neural parametrization of the dual, trained as a physics-informed neural network (PINN), that encodes Harrison's equivalent-workload formulation (see Harrison 2000) matches the strong reinforcement-learning baseline of Dai and Gluzman (2022); numerically conditioning the fit of a sub-dominant transverse correction then improves on both that baseline and the best priority heuristic. On the input-queued switch, a graph-attention parametrization of the dual improves on the strongest matching heuristic we are aware of, with a parameter count that does not grow with the switch size.

The Taylor diagram is used to evaluate and compare predictive models with observed data and has many applications in climate and environmental sciences. Three statistics of interest–the centered root-mean-squared error, standard deviation, and the product-moment correlation coefficient–are related by the law of cosines; Taylor diagrams leverage this fact to allow visualization of all three statistics in a two-dimensional plot without any loss of information. In this work, we present a novel model evaluation tool–the Wasserstein-Taylor diagram–formed from integrating the Wasserstein distance from optimal transport into a Taylor diagram framework. This tool is built upon the fact that the law of cosines still holds in the Taylor diagram if the centered root-mean-squared error and product-moment correlation coefficient are replaced by the centered 2-Wasserstein distance and quantile correlation coefficient, respectively. The advantage of this Wasserstein-Taylor diagram is that one can use the distribution perspective afforded by the Wasserstein distance to compare observed and predicted data sets of different sizes and even to compare statistical models with only observed data. We further show that the quantile correlation coefficient on empirical measures converges to the quantile correlation coefficient on the sampled measures.

For the neutral periodic one-dimensional Riesz gas with pair potential locally $-|x|^a$, $0<a<1$, we prove a particle-displacement variance bound of order $\beta^{-1}$, uniformly in the number of particles. Log-concavity also gives exponential displacement tails. Every stationary periodic thermodynamic limit is simple, has intensity one, and admits a stationary ordered matching to the unit lattice with the same bounds. Each limit satisfies the canonical Gibbs equations for the full-line interaction, with an ordinary symmetric spatial principal value for the exterior potential. The matching implies uniformly bounded interval number variance and a positive limiting second moment of the reciprocal-lattice Fourier average at sufficiently low temperature. The main estimate compares the inverse random Hessian, through deterministic electrical flows, to a transient long-range network.

For a ferromagnetic Ising model on a graph of maximum degree $\Delta\ge3$, we prove a bound of order $\sqrt n$ on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by $\beta$ and $\varepsilon=((\Delta-1)\tanh\beta-1)_+$, the bound is $C(\sqrt n+n\varepsilon)$. The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window $\varepsilon=O(\sqrt{\log n/n})$, with the polynomial degree depending on the window size.