Papers

Detection of geometry in random graphs and matrices has been extensively studied in recent years. For testing a dense random geometric graph on $n$ vertices with latent points in dimension $d$ against an Erdős–Rényi graph, the detection threshold $d\asymp n^3$ is well known since Bubeck, Ding, Eldan, and Rácz (2016). We sharpen this result by establishing the optimal asymptotic testing error in the critical regime $d/n^3\to\lambda\in(0,\infty)$. The log-likelihood ratio is shown to be asymptotically normal and dominated by the signed triangle count. Moreover, we consider testing the rank statistics of inner products between latent points against uniformly random ranks. We show that ranks preserve all asymptotic testing information available in the full matrix models, and a cubic statistic based on rank statistics attains the optimal asymptotic error. This then yields an asymptotically optimal test for random kernel matrices with a possibly unknown, strictly monotone kernel. The proofs combine the classical framework of Davies (1985) and Le Cam and Yang (1988) for asymptotic normality under information loss with conditional limit theorems for the models in consideration.

The 1-Meixner random vectors consist of $d$ real-valued random variables, having finite moments of all orders, for which the commutators between their joint semi-quantum operators and the multiplication operators that they generate are linear combinations of these multiplication operators and the identity operator. This property of random variables imposes a system of first-order partial differential equations on their joint Laplace transform on a neighborhood of $0$ in a Euclidean $d$-space. We provide explicit descriptions of all solutions of the system just mentioned in every dimension $d$.

We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle $S_a = a \otimes 1 + \sum_j A_j \otimes s_j$ with deterministic mean $a$. For every real $z$, $\log \Delta(S_a - z) = -1/2 + \inf_b F_z(b)$, where the infimum is over positive definite matrices $b$ such that the map $X \mapsto bXb - \sum_j A_j X A_j$ is positive semidefinite, and $F_z$ is an explicit geodesically convex function. Both sides equal $-\infty$ exactly when $z$ is an atom of the spectral measure. The formula was proposed, without proof, in a recent post of Speicher. In the centered case $a=0$ it gives a new proof of the Mai-Speicher capacity formula, and by hermitization it yields a finite-dimensional variational formula for the logarithmic potential of the Brown measure of an arbitrary matrix-valued semicircular element. The proof identifies the determinant with the value of the Dyson action at its critical point and rewrites this critical-point problem as a saddle problem in three matrix variables. For a spectral parameter $z + i\varepsilon$ in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor $b_\varepsilon$ in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting $\varepsilon \to 0$ in the optimal values, with no compactness assumption on the minimizers. At points where the boundary value of the Cauchy transform is nondegenerate, the scales $b_\varepsilon$ converge and the infimum is attained at their limit; this minimizer lies on the boundary of the feasible set when $z$ is in the support of the spectral measure and in its interior when $z$ is in the resolvent set.

The distribution function $F(y)=\Pr(S^2\le y)$ of the sample variance $S^2$ of four independent observations is elementary on its entire support for every polynomial parent density on $[0,1]$ (without loss of generality), and we give it in finite closed form for the uniform, Beta$(2,1)$ and Beta$(2,2)$ parents. The support splits at $y = 1/6$, $2/9$ and $1/4$, with the ceiling at $1/3$, all fixed by geometry: in coordinates aligned with the cube diagonal the event $S^2\le y$ is a ball of radius $\sqrt{3y}$, the cube projects to a rhombic dodecahedron, and these are the levels at which the ball reaches its faces, edges and two kinds of vertices. The divergence theorem reduces every moment of the clipped ball to integrals along circular arcs, so each regime closes in algebraic functions of $\sqrt{y}$ and inverse trigonometric functions. For the uniform parent the only non-obvious constant is $K_g = 12\sqrt2\arctan\sqrt2$, proportional to the mean width of the cuboctahedron; the Beta constants are rational expressions in $\pi$, $\sqrt2$, $\sqrt3$ and $\arctan\sqrt2$, as in the uniform case. At $n = 4$ the normal-theory $\chi^2$ recipe used in acceptance testing misplaces the $99$th percentile of $S^2$ by $50\%$ for the uniform parent. For three observations, polynomial parents also give elementary closed forms (arXiv:2609.25333), but two elementary cases in a row do not guarantee a third, and we make no claim beyond $n = 4$.

We continue the study of weak universality for the fractional hyperbolic $\Phi^4_3$-model initiated by Tzvetkov, Wang, and the author (2023). In this work, we establish weak universality in a regime where the associated Gibbs measure is mutually singular with respect to the underlying Gaussian measure. We first investigate the weak convergence of the Gibbs measures associated with macroscopic models to the fractional $\Phi^4_3$-measure constructed by Tzvetkov, Wang, and the author (2023). We then obtain a structured representation for these Gibbs measures via the stochastic quantization method. Finally, we prove dynamical weak universality for the fractional hyperbolic $\Phi^4_3$-model by using an extended second order expansion and exploiting the invariance of the Gibbs measures.

We provide a new rearrangement in Gauss space transforming symmetric sets while decreasing their Gaussian perimeter. We use this rearrangement to analyze the minimizers of the Gaussian perimeter among symmetric sets.

This paper studies the last-coverage vertex of a nearest-neighbour random walk on a cycle: the vertex most likely to be the final one discovered. We show that the visited vertices always form a single contiguous arc of the ring, a property we call the Connected Arc Property. This lets the walk be replaced by a two-state time-inhomogeneous Markov chain, which we call the interval-expansion representation, in which each expansion step has a probability given by the classical gambler's ruin formula. Within this framework we prove that the last-coverage probability of node k in an n-node cycle is (1-r)r^(n-1-k)/(1-r^(n-1)), where r = (1-p)/p and p is the clockwise transition probability. As p approaches 1/2, we recover the uniform distribution, giving an independent derivation of a classical result. For a walk with a deterministic time-varying bias, we give an exact series representation for the last-coverage probabilities, together with a finite truncation that serves as a computable, error-bounded approximation. All results are validated by Monte Carlo simulation, and we run experiments on sinusoidal time-varying bias. Finally, we compare generalized Petersen graphs with Erdos-Renyi graphs of matching size, finding that structural symmetry is associated with more uniform last-coverage distributions.

In this paper, we study finite-velocity random motions subject to a killing mechanism acting at direction changes: at each change of direction, the motion is killed with a fixed probability and remains at its current position thereafter. We first consider the one-dimensional case of a standard telegraph process. We derive the exact distribution of the position at the killing time and show that it is Laplace distributed. We also characterize the law of the process conditional on survival up to a fixed time, proving that it coincides with that of a standard telegraph process driven by a thinned Poisson process with reduced switching intensity. We further derive the full time-dependent distribution of the killed process, including both its absolutely continuous and singular components. We then extend the analysis to a planar finite-velocity random motion with infinitely many possible directions. In this setting, we derive the distribution of the killed process and obtain an explicit representation of its density in terms of modified Bessel functions and a series involving Bessel polynomials.

This article is concerned with the problem of nonlinear (stochastic) filter stability for a discrete time hidden Markov model (HMM) with finite state and observation spaces. For this purpose, a comprehensive duality theory is developed whereby the dual control system is a backward stochastic difference equation (BSDE). The duality is expressed in two inter-related forms. In its variational form, the minimum mean squared error of optimal filtering equals the value of a dual optimal control problem. In its structural form, controllability (resp. stabilizability) of the BSDE and observability (resp. detectability) of the HMM are dual properties. The dual optimal control problem supplies the Lyapunov function and the dissipation: the former is the value function (a variance), the latter the optimal running cost accumulated over the horizon. The main result is an invariance principle: under any one of three model properties—ergodicity of the Markov matrix, or observability or detectability of the HMM—zero dissipation over a finite horizon forces the Lyapunov function to vanish. A converse is also obtained: detectability is necessary. Two applications are given. The first is filter stability in chi-squared divergence: the invariance principle is applied in the manner of LaSalle, to limits of trajectories along which the dissipation vanishes asymptotically, the role of the model properties being to render the zero-dissipation set trivial. The second is a quantitative form of the invariance principle, through a Poincare constant, from which convergence rates follow. Both the model properties that govern stability and the Lyapunov function that certifies it are thus obtained from a dual optimal control problem, as in the classical theory for the linear Gaussian model.

Fix $0<\kappa<8$ and put $d=1+\kappa/8$. Recent work by OpenAI identifies an explicit exact Hausdorff gauge \( h_\kappa(r)=r^d\bigl(\log\log(1/r)\bigr)^{(2-d)/2} \) for sufficiently small $r$ for chordal $\mathrm{SLE}_\kappa$: the corresponding Hausdorff measure is almost surely positive and finite on every segment $\gamma([s,t])$ with $0<s<t<\infty$, and has finite expectation in bounded subsets of the plane. We show that this Hausdorff measure is a deterministic positive constant multiple of the occupation measure (natural length) constructed by Lawler and Rezaei.

We show that uniqueness may fail for one-dimensional McKean–Vlasov stochastic differential equations with Hölder continuous diffusion coefficients. More precisely, for every \(\alpha\in(1/2,1)\), we construct an equation with a diffusion coefficient that is \(\alpha\)-Hölder continuous in the state variable, Lipschitz continuous in the measure variable, and of linear growth, for which uniqueness in law and pathwise uniqueness fail. The construction relies on a spreading estimate together with a Schauder fixed-point argument.

We present DisCo algebra—a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not—the information loss is exactly the conditional KL divergence between the true conditional distribution and the conditionally independent product. This asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries.

We study the Gibbs measure of $H(h) = \Re\sum_{p^{j} \leq X}U_{p}^{j}/(j p^{j(\sigma + i h)})$, a random field modeling large values of the log-modulus of the Riemann zeta function on and near the critical line, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. Writing $t = \log \log T$, the cutoff $X$ and real part $\sigma$ may vary with $T$ subject to $T^{1/t^{\Phi}} \leq X \leq T^{1/t^{\phi}}$ and $0 \leq \sigma - 1/2 \leq C_{\sigma}t^{\Phi}/\log T$, for fixed $\Phi \geq \phi > 1/2$ and $C_{\sigma} > 0$. On $I_{\theta} = [ - (\log T)^{\theta},(\log T)^{\theta}]$, define the Gibbs measure $G_{\beta,\sigma,T}^{\mathrm{H}}(\mathrm{d}h) := e^{\beta H(h)}\mathrm{d}h/\int_{I_{\theta}}e^{\beta H(h')}\mathrm{d}h'$. For every fixed $\theta > - 1$ and inverse temperature $\beta > 0$, the limiting joint distributions of correlations (overlaps) among points sampled from this measure are determined after averaging over the Haar phases. At and below $\beta_{c}(\theta) = 2\sqrt{1 + (\theta \vee 0)}$, the limiting two-overlap distribution is concentrated at $q_{0} = |\theta \wedge 0|$. Above this threshold, it assigns masses $\beta_{c}(\theta)/\beta$ and $1 - \beta_{c}(\theta)/\beta$ to $q_{0}$ and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter $\beta_{c}(\theta)/\beta$. Approximate Ghirlanda-Guerra identities are shown to hold uniformly for bounded deterministic functions of the sampled positions. This paper is the first in a series devoted to extending these results to $\log|\zeta|$ near the critical line.

We establish dimension-free, pointwise optimal lower-deviation bounds for convex functions under spherically symmetric Cauchy-type measures. These bounds use either mean-variance normalization or moments about the median, the latter covering the Cauchy case where the mean and variance are unavailable. By approximation with a parametrized family the Cauchy-type measures, our results recover the Gaussian setting of Paouris and Valettas \cite{PaourisValettas18} and strengthen their variance-sensitive estimate to a sharp bound, with a complete equality characterization. This goes beyond the partial improvements obtained in subsequent work. We also prove an Ehrhard-type inequality for arbitrary convex sets under these Cauchy-type measures in every dimension and characterize all equality cases. This resolves a conjecture posed by Byczkowski, Małecki and Żak \cite{ByczkowskiMaleckiZak22}. The proof follows from a variational argument combined with the joint concavity of fiber rearrangement. The geometric inequality reduces distributions of convex observables to one-dimensional convex transforms, yielding the sharp lower-deviation bounds.

The $(d+1)$-type $\Lambda$-Wright–Fisher process with polynomial drift is a Lévy-type process on the $d$-dimensional unit simplex whose generator entails a particular jump and diffusive component, as well as a polynomial drift component. In this work, the process is connected to a (Bernstein) dual process, which admits an interpretation as an urn model with coloured balls. Exploiting this duality, we identify various parameter regimes: one in which the $\Lambda$-Wright–Fisher process is absorbed almost surely at a vertex of the simplex, and others in which it admits a unique nontrivial ergodic probability measure. The results include verifiable criteria for absorption versus persistence, as well as representations of absorption and stationary measures. Moreover, the duality relation is exploited to devise and implement a simulation-based approach for estimating moments of the $\Lambda$-Wright–Fisher process.

In the classical occupancy problem, $B$ balls are placed independently and uniformly into $d$ boxes. We study a graph analogue of this model. Let $G$ be a directed $r$-regular graph on $d$ vertices and fix a vertex $i$. Consider all paths of length $L$ starting from $i$, and let $N_L(i,j)$ denote the number of such paths ending at a vertex $j$. Thus the vertices play the role of boxes and the $r^L$ paths play the role of balls. In general, these occupancies depend strongly on the geometry of $G$. We show that inserting independent uniform permutations between successive layers restores the classical Poisson behavior at the empirical level. Although the path endpoints remain strongly dependent, their empirical occupancy distribution has the same limiting Poisson law as independent balls thrown into boxes. More precisely, at the critical scale $r^L/d\to\lambda\in(0,\infty)$, provided $rL/d \to 0$, the empirical path multiplicity distribution converges in Wasserstein distance to $\operatorname{Pois}(\lambda)$, independently of the geometry of the underlying graphs. We also prove a local Poisson limit theorem and a Berry-Esseen bound for the empirical distribution functions. These estimates determine the full total-variation transition of the normalized path multiplicities toward uniformity. For arbitrary doubly stochastic kernels, we further establish exact covariance and Frobenius norm identities, together with an exact contraction formula for the chi-square divergence. As applications, we study synchronous opinion averaging, obtaining an explicit cutoff profile for equal-size groups and quantitative convergence to consensus for heterogeneous group sizes.

We study upper tail large deviations for the first passage time in the frog model on $\mathbb Z^d$, $d\ge 2$. Let $\mathrm{T}(0,x)$ denote the first time at which the infection, started from the origin, reaches $x$, and let $\mu$ be the associated time constant. We prove that, for any Euclidean unit vector $v\in\mathbb R^d$ and $\xi>0$, \[ \lim_{n\to\infty} -\frac{1}{\mathfrak{s}_d(n)} \log \mathbb{P}\bigl(\mathrm{T}(0,\lfloor nv\rfloor) \ge (\mu(v)+\xi)n\bigr) =\mathfrak{r}_d\xi, \] where $\mathfrak s_2(n)=n/\log n$ and $\mathfrak r_2=\pi$ for $d=2$; $\mathfrak s_d(n)=n$ and $\mathfrak r_d$ is the escape probability of simple random walk for $d\ge3$. Moreover, on the upper tail large deviation event, we show that a certain slowdown mechanism occurs near the endpoint. To rule out other mechanisms, we analyze the growth of the infection near the starting point using a modified exploration process, and investigate its propagation through the intermediate region using the renormalization and covering arguments.

We study a forward-backward McKean-Vlasov system with jumps whose backward equation contains a source obtained by evaluating the minimal-norm selection of a maximal monotone operator along the forward state. This selection changes branch across an interface that moves with the state law, the control, and an imprecisely known parameter described by fuzzy cuts. Assuming normal non-degeneracy and finite-statistics dependence of the interface on the law, we prove that the first and second moments of the time spent within a given distance of the interface are bounded, respectively, by a constant times that distance and by a constant times its square. The second-moment estimate makes it possible to integrate the source error before squaring it, which yields a Yosida error of first order in the regularization parameter in the natural solution norm; a first-moment argument gives only order one half. An explicit example shows that the first-order rate is sharp. For compact classes of admissible selections, the corresponding solutions form the cuts of a fuzzy solution set. These cuts are compact, stable with respect to the fuzzy level and the data, and approximated at the same first-order rate in Hausdorff distance. All estimates are uniform over a non-dominated family of probability measures representing volatility uncertainty. The quantitative results are obtained for separable quadratic potentials with finitely many affine branches.

We study random contingency tables obtained by conditioning a matrix of independent Poisson entries on its row and column sums. We identify such a table with the block counts of a random permutation drawn from a Gibbs measure of Mallows type, an exact coupling whose normalizing constants recover Barvinok's permanent identity. We use this correspondence to study the table at three scales. For finite dimensions, a transference principle compares the table with a matrix of independent Poisson entries whose means form the static Schrödinger bridge, the solution of a discrete entropic optimal transport problem. As the dimensions grow and the margins and prior means converge, the normalized histogram of the table converges to a limiting Schrödinger bridge at a quantitative rate. The fluctuations of the table around the bridge are characterized in two complementary ways. Linear statistics satisfy a central limit theorem for bounded test functions, with limiting variance equal to the squared $L^2$-distance, under the bridge, of the test function to the subspace of additively separable functions; as a process they converge to white noise with intensity given by the bridge, conditioned to have vanishing marginals. Spectrally, the limiting spectral distribution of the centered table is governed by a Dyson equation whose variance profile is the density of the bridge. The key ingredient is an asymptotic formula for the partition function of the Gibbs measure in terms of a Fredholm determinant given by the canonical correlations of the bridge. For Mallows-type random permutations, our results prove Pal's conjecture on the limiting partition function and Mukherjee's conjecture on the central limit theorem for linear statistics, both in greater generality.

We consider spin-coupled Grover walks on hypercubes within the framework of quantum Bernoulli noises (QBN). We identify parity-dependent harmonic subspaces generated by the central Hamming layers: in even dimension they give a fixed-point subspace, whereas in odd dimension the walk reduces on the harmonic subspace to an explicit eight-dimensional reduced dynamics. Using symmetric-group representation theory, we characterize the corresponding isotypic structure and determine the reduced spectrum. The harmonic reduction also yields explicit long-time probability behavior and shows that the resulting states may occupy an asymptotically vanishing fraction of the hypercube while still spanning its full Hamming diameter.

We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \infty)^2$ governed by the Matsumoto–Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $d\mu = \frac{dx \, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.

In this work, we develop and analyze a splitting mixed finite element method, combined with a fully implicit Euler time discretization, for a stochastic Keller–Segel system with general polynomial logistic growth and multiplicative noise. The proposed splitting formulation allows the use of standard finite element spaces without requiring the LBB inf–sup condition associated with traditional mixed finite element methods and, at each time step, avoids the need for solving a coupled nonlinear system. We first establish the well-posedness of the variational solution and derive new high-moment, higher-order regularity, and exponential stability estimates. These estimates provide the key analytical foundation for the strong convergence analysis of the numerical method. A principal contribution of this work is the derivation of error estimates directly in full expectation, without the use of localization arguments. By combining the new stability and regularity estimates with a stochastic Gronwall framework, we overcome the difficulties arising from the interaction between the nonlinear chemotactic coupling and the multiplicative noise. As a result, we establish strong convergence in high-moment norms and obtain optimal convergence rates with respect to both the temporal and spatial discretizations. In particular, the proposed method achieves the optimal rates of order $k^{\frac{1}{2}}$ in time and $h$ in space in the relevant $L^2$- and $H^1$-based error norms. These results provide, to our knowledge, the first optimal strong convergence analysis in full expectation for a splitting mixed finite element approximation of this class of stochastic Keller–Segel systems.

We establish variance upper bounds for the minimum perfect matching problem in general metric spaces under a geometric partition condition. As a principal application, let $H_n$ be the minimum perfect matching cost of $n$ independent uniform points in $[0,1]^d$, where $n$ is even and $d\ge2$. We prove $$\operatorname{Var}(H_n)\le C d n^{1-2/d},$$ where $C$ is independent of $n$ and $d$. This removes the logarithmic losses in earlier upper bounds in every dimension (Talagrand, 1995); in particular, the variance remains bounded in dimension two, answering an open question (Rhee, 1994). The proof relies on constructing a rapidly mixing Markov process with a multinomial stationary distribution.

We prove a central limit theorem for the quadratic matching cost between the uniform measure and its empirical counterpart on the flat torus in dimensions $d\ge5$. Let $\mu$ be Haar probability measure on $\mathbb{T}^d$ and let $\mu_n$ be the empirical measure of $n$ independent samples from $\mu$. Then $$ \begin{aligned} \frac{W_2^2(\mu_n,\mu)-\mathbb{E}W_2^2(\mu_n,\mu)} {\sqrt{\operatorname{Var}(W_2^2(\mu_n,\mu))}} &\Longrightarrow N(0,1),\\ c_dn^{-1-4/d} \le \operatorname{Var}(W_2^2(\mu_n,\mu)) &\le C_dn^{-1-4/d}, \end{aligned} $$ for positive constants $c_d,C_d$ depending only on $d$ and all sufficiently large $n$. The proof introduces a random-walk approach to matching fluctuations. We construct a random walk that jumps between neighboring cells of the optimal partition and use it to describe how the matching cost responds to changes in the sample. Estimates for this walk control the effect of replacing one sample point and the interaction between two replacements. These bounds yield the variance estimates and allow us to apply a martingale central limit theorem.

We study the geometry of semi-infinite geodesics in the Airy line ensemble. We establish sharp estimates for disjointness probabilities involving geodesics with nearby directions and endpoints in the top line of the Airy line ensemble. Moreover, we obtain a power law upper bound on the coalescence depth of two semi-infinite geodesics with same asymptotic direction and establish exponential concentration estimates for geodesic jump times more generally. We also establish pathwise Hölder $1/2^-$-continuity of Busemann profiles associated to semi-infinite geodesics with common asymptotic directions (in terms of the asymptotic direction). We characterize exceptional behaviour of semi-infinite geodesics through their Busemann profiles and establish geometric properties of the associated sets of non-coalescence and non-uniqueness. Finally, we determine the Hausdorff dimensions of collections of semi-infinite geodesics: collections with a fixed top-line endpoint and arbitrary positive asymptotic direction almost surely have Hausdorff dimension $1$, while collections with a fixed positive direction and arbitrary endpoint almost surely have Hausdorff dimension $3/2$.

Generative modelling involves two complementary tasks: learning a probabilistic approximation of an unknown data distribution from observations, and designing a tractable mechanism for sampling from this approximation. We present a stochastic-analysis perspective on score-based generative models, connecting ideas from classical sampling theory, score estimation, and diffusion time reversal. We first explain how score functions link Langevin sampling with score matching, and how Gaussian smoothing leads to denoising score matching and multiscale sampling methods. We then emphasize the distinction between dynamics preserving a fixed distribution and dynamics transporting a prescribed family of marginals. This motivates a path-space treatment of diffusion time reversal. Finally, we connect the ideal reverse diffusion with practical generative algorithms.

In the abelian sandpile model on $\mathbb{Z}^d$, a site with at least $2d$ grains topples by sending one grain to each neighbor. In a box $[-N,N]^d$ whose boundary absorbs grains, adding grains at random sites and toppling defines a Markov chain. Athreya and Járai (2004) constructed the stationary state on $\mathbb{Z}^d$ as the limit of its stationary distributions as $N\to\infty$. They asked whether, when a grain is added at the origin of a sandpile in the stationary state, the resulting avalanche of topplings is finite. Járai and Redig (2008) proved that the answer is yes for $d\geq 3$, using the transience of random walk. We prove, using a different method, that avalanches are also finite for $d=2$.

We study the convergence of Markov chains in the geometry induced by Hilbertian embeddings associated with positive definite kernels, also called Maximum Mean Discrepancy (MMD). Under a contraction assumption in the kernel-induced geometry, and a standard spectral gap assumption, we establish non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure. The obtained bounds separate the roles of the two conditions: the spectral gap determines the effective sample size in the leading term, while the kernel contraction enters only through a linear correction. As an application, we study one-step prediction of dynamical systems from a single trajectory, deriving fast learning rates for a nonlinear least-squares estimator. The rates highlight the interplay between mixing properties in the kernel-induced geometry and the classical source and capacity conditions used in the analysis of least-squares.

Consider a queueing system operated by Bob, which is intended to serve only jobs generated by a legitimate user, Willie. Willie's jobs arrive according to a Poisson process, and the service times are exponentially distributed. Bob seeks to covertly serve additional jobs arriving according to an independent Poisson process, without Willie detecting their presence. As the legitimate user, Willie is provided with certain statistics of each busy period by Bob. In particular, we consider two statistics: (i) the number of jobs served during each busy period and (ii) the duration of each busy period. We establish that, under either statistic, the maximum expected number of additional jobs that Bob can covertly serve obeys a square-root law. Specifically, over n busy periods, Bob can covertly serve additional jobs in the order of square root of n, whereas any scaling which is above square root of n enables Willie to asymptotically detect the presence of these additional jobs with vanishing error probability. Furthermore, we characterize the scaling constant associated with the square-root law, which we refer to as the covert queueing constant, for each of the two statistics. We compare the scaling constants corresponding to the two statistics and argue that the scaling constant under statistic (i) is no greater than that under statistic (ii).

We introduce a general class of interacting reinforced stochastic systems evolving on a network. The dynamics are governed by two distinct interaction structures: a sampling network, which determines how information is acquired, and a reinforcement network, which determines how the sampled information influences future evolution. Processes generate Bernoulli observations through preferential, de-preferential, or mixed sampling, while observations are transmitted through an edge-dependent affine reinforcement mechanism. We formulate the dynamics as a stochastic approximation scheme and study their asymptotic behaviour. We establish sufficient conditions for almost sure convergence to a deterministic limit and, when the sampling and reinforcement graphs coincide, derive graph-theoretic criteria involving odd cycles, mixed-sampling vertices, and subcritical reinforcement. Under additional structural assumptions, we establish almost sure convergence and partial synchronization in a random-limit regime. In the deterministic regime, we also prove a law of the iterated logarithm and a central limit theorem under suitable spectral and regularity conditions. Finally, we show how the framework encompasses interacting urn models with edge-dependent reinforcement, urns with memory, and random walks with memory.