Papers

How large of a minimum degree does an $n$-vertex graph need before we are sure that it contains a perfect matching? Dirac's theorem states that a graph on an even number of vertices in which each vertex has degree at least $n/2$ has this property. In this short expository note, intended to be used in the classroom, we discuss how this statement generalizes to hypergraphs. In particular, we highlight an elegant connection between fractional perfect matchings in hypergraphs and a probabilistic inequality about nonnegative random variables, which was conjectured by Feige. We also present a very short self-contained proof of Feige's conjecture.

For each real $b$, exact upper and lower bounds on the probability $\mathsf P(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf E X=0$ and $\mathsf E X^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf P(X\ge b)\le C e^{-b}$ for all real $b\ge0$. Exact upper and lower bounds on the corresponding exponential moments (Proposition 9.4) and explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s (Propositions 9.2 and 9.3) are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.

We propose a neural algorithm for sampling from distributions specified by unnormalized Boltzmann densities. Our approach is based on the Jordan–Kinderlehrer–Otto scheme for the Kullback–Leibler divergence in the Wasserstein–Fisher–Rao geometry (WFR JKO scheme). Our contributions are twofold. First, we prove that, for any fixed step size, the exact WFR JKO iterates converge exponentially fast to the target as the number of iterations tends to infinity. Notably, this result requires no structural assumptions on the target, such as log-concavity or a logarithmic Sobolev inequality. Second, we develop a neural implementation of the WFR JKO scheme that parametrizes its transport and reaction components using reweighted normalizing flows. Numerical experiments on challenging multimodal targets demonstrate the promising performance of the proposed method.

Entropy-regularized optimal transport (EOT) has become a key tool in optimal transport, providing both theoretical insight and efficient computation via the Sinkhorn algorithm. We study the first-order \(\Gamma\)-expansion of EOT as the regularization parameter \(\varepsilon\) vanishes, by rewriting EOT as the minimization of a relative entropy with respect to a Gibbs measure \(\gibbs\) associated with the duality gap energy. Using a mild extension of the Laplace method, we prove the convergence of the Gibbs measures and derive this \(\Gamma\)-expansion for non-degenerate ground costs, under suitable assumptions on the duality gap, the contact set and the marginals. This yields at the same time an entropic selection principle and an expansion of the EOT cost up to \(o(\varepsilon)\) for this class of ground costs, generalizing existing results.

We establish general conditions for the weak convergence of regular one-dimensional diffusion processes when their scale functions, speed measures, state spaces and boundary conditions may substantially vary. Our approach is based on the classical Itô–McKean representation of regular diffusions via time changes and scale transformations of a single Brownian motion. This reduces many stochastic convergence problems to deterministic questions concerning the convergence of integrals, monotone functions, compositions, and generalized inverses. As a main application, we investigate various homogenization regimes for diffusions in media with closely spaced semipermeable membranes, where the resulting limiting processes include not only reflected and skew Brownian motions, but also skew-sticky Brownian motions and snapping-out diffusions. This yields a representation of a snapping-out diffusion as a function of a single Brownian motion, without introducing an auxiliary sequence of independent exponential random variables.

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1<p<\infty$ whenever $f\mapsto T_{t_0}(\mathbf 1_Ef)$ is compact on $L^2(\mu)$ for some $t_0>0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein–Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\to \infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\ge2.$

An optimal stopping problem is monotone when its stopping region is preserved under subsequent transitions, so that the one-step look-ahead stopping time is optimal. We develop a finite-horizon discrete-time theory of multiple stopping that separates general recursive dynamic programming from this stronger monotone structure. Extending the classical martingale-system approach, we construct a recursively defined optimal vector of stopping times without assuming monotonicity. If monotonicity holds at every exercise level, this vector coincides with the recursive vector of one-step look-ahead stopping times. A bounded counterexample shows that monotonicity of the original single-stopping problem alone is insufficient, whereas monotonicity propagates to every level when the level-one one-step look-ahead function is deterministic. We apply the theory to multiple-selection last-success problems. For the classical independent odds setting, a continuous-time Poisson embedding gives an alternative proof of the multiple-stopping lower bound and clarifies the associated threshold constants. We also examine random horizons, Markov-dependent trials, and an unknown success probability

This article generalizes the de-initialization framework, proposed by (Roberts and Rosenthal, 2001) for total variation distance, to Wasserstein distances. In essence, de-initialization captures the phenomenon that in the presence of certain structural features, the convergence behaviour of a Markov chain can be reduced to that of a "simpler" stochastic process. Our results allow to measure this in terms of a general Wasserstein distance. We apply them to analyse the Wasserstein convergence behaviour of Gibbs sampling, linchpin variable sampling and slice sampling.

Let $C_1,\ldots,C_k$ be the convex hulls of independent partial-sum processes in $\mathbb{R}^d$ whose increments are exchangeable within each process. We express the expected mixed volume $\mathbb{E} V_d(C_1[m_1],\ldots,C_k[m_k])$, $m_1+\cdots+m_k=d$, through mean absolute determinants of disjoint block sums of the increments; no general-position assumption is needed. For random walks with i.i.d. integrable increments the block sums are independent, and the formula extends the expected-volume formula of Barndorff-Nielsen and Baxter and of Vysotsky and Zaporozhets to mixed volumes. We then prove a continuous-time counterpart: for independent Lévy processes with finite first moments, the expected mixed volume of the closed convex hulls of their paths is an explicit integral of mean absolute determinants over a product of simplices. For symmetric stable processes the integral is evaluated in terms of the associated zonoids. As a geometric application, we compute the mean mixed volume of random projections of mutually orthogonal canonical orthoschemes.

A centrally excited random walk on ${\mathbb Z}^d$ moves like simple random walk, except that its first step from each site has a drift of fixed size toward the origin. Kozma (2007) conjectured that after $n$ steps the visited set approximates a ball with radius of order $n^{ 1/(d+1)}$. We prove this in every dimension $d\geq2$ and show that the rescaled numbers of visits converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that a drift opposite to a subgradient of a norm produces the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most $ n^{1/6}$ times a power of $\log n$, and the exponent $1/6$ cannot be lowered.

We establish sharp two-point function asymptotics for the two dimensional Villain model at sufficiently low temperatures, confirming a central prediction in the Berezinskii-Kosterlitz-Thouless theory. We characterise the power law exponent through the renormalisation group flow of a marginal line defect observable in the dual high-temperature Discrete Gaussian model.

Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the kernel parameters are routinely estimated by the maximum likelihood estimators (MLEs), as implemented by standard GP software. Despite this widespread use, the asymptotic behavior of the MLEs remains poorly understood under fixed-domain asymptotics, even for the RBF kernel. The main difficulty arises from the increasingly strong dependence among densely sampled observations and the nonlinear dependence of the covariance matrix on the kernel parameters. In this paper, we address this gap by providing, to the best of our knowledge, the first complete asymptotic characterization of the joint MLE of the spatial variance, lengthscale, and nugget variance under fixed-domain asymptotics. We establish consistency, derive convergence rates for all three parameters, prove joint asymptotic normality, and show that these rates are minimax optimal.

Ferromagnetic spin models on random regular graphs, such as the Ising and Potts models, have been studied extensively. Several equivalent variational and recursive representations of their limiting pressure are known, and these representations can be used to establish agreement between the quenched and annealed pressures. We give a concise overview of these representations and highlight the connections among the different formulations. We then derive analogous representations for the exponential growth rate of the second moment of the partition function, and prove that it has the same order as the square of the annealed pressure, up to a subexponential error. Combined with concentration of the quenched pressure, this identity yields a transparent second-moment proof of quenched–annealed agreement on random regular graphs. By bringing these representations and arguments together in a unified presentation, we aim to provide an accessible entry point to the literature on ferromagnetic spin models on random regular graphs.

A theoretical understanding of multi-head self-attention is fundamental to the study of modern neural networks. Using random matrix theory, we establish Gaussian equivalence for multi-head self-attention: replacing softmax attention with rescaled scores plus Gaussian noise preserves the limiting spectral law of the centered output. This equivalence also covers value and output projections that depend on the keys. The resulting laws separate the effects of head allocation and projection widths, and distinguish spectrum-preserving across-head sharing from within-head key–value dependence.

In this paper, we establish two nonasymptotic Berry–Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor $d^{1/4}$ in the contribution of the independent sum, where $d$ is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of independent random vectors, it removes the logarithmic factor from an existing fourth moment bound without imposing additional moment assumptions. As applications, we apply these results to Polyak–Ruppert averaging for nonsmooth stochastic approximation, temporal difference learning with linear function approximation, and multivariate $U$-statistics. The resulting bounds provide explicit Gaussian approximation errors and sufficient conditions under which these errors converge to zero as the dimension grows with the sample size.

We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals $1/n$, where $n$ is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved $5$-cube, which mixes uniformly at time $\pi/4$. With the order $4\theta^2$ read literally, its parity condition also excludes the Clebsch graph and includes the parameters $(36,14,4,6)$, for which no time law gives uniform average mixing.

We study sampling from the Gibbs distribution of the Sherrington-Kirkpatrick (SK) model with Glauber dynamics. For every fixed inverse temperature $0\leq\beta<1$ and every fixed $M,D>0$, we give a polynomial-time simulated annealing algorithm whose output distribution is within $n^{-M}$ total-variation distance of the Gibbs distribution, with probability at least $1-n^{-D}$ over the interaction matrix. The algorithm starts from uniform product spins and uses Glauber dynamics along an increasing inverse-temperature schedule. The same approach also yields partition-function estimates with relative error $n^{-M}$. The key ingredient is a quantitative weak Poincaré inequality. Building on the stochastic-localization approach to weak Poincaré inequalities [Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026], we use Gaussian isoperimetry to directly compare exact stochastic localization paths. Our comparison tolerates failures of local Lipschitz continuity of the posterior mean by controlling their accumulated cost. The local Lipschitzness comes from approximating the posterior means by stationary points of the Thouless-Anderson-Palmer (TAP) free energy in locally strongly convex regions located by approximate message passing, a paradigm introduced by algorithmic stochastic localization [El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]. The approximation errors are roughly characterized by the validity of the TAP gradient equations, but in a different coordinate. To show that the errors rarely accumulate too much, we need a strong concentration control. A direct argument would require solving a variant of an open problem by Talagrand [2010, Research Problem 1.7.9]. We bypass this open problem with an intermediate conditioning step and transfer the control back by establishing the stability of the TAP gradient under deletion of coordinates.

We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator $h^2\Delta+V(x)$ that is perturbed by a smooth potential $h^\alpha V_\omega$ with $\alpha\in (0,1)$, and $V_\omega$ is a random potential that decorrelates on distances $h^\beta$, with $0<\beta<2\alpha$. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.

We develop a non-asymptotic theory for nonparametric drift estimation in discretely observed multi-dimensional Lévy-driven stochastic differential equations using sparse ReLU neural networks. Under exponential \(\beta\)-mixing and an exponential-tail condition on the Lévy measure, we establish an oracle inequality for the least-squares estimator. The jump component changes both the martingale structure and the concentration regime relative to diffusion models: continuous martingales are replaced by discontinuous martingales, while the relevant empirical-process fluctuations are sub-exponential rather than sub-Gaussian. We handle these difficulties without truncating the observed increments, using an exponential-supermartingale argument for compensated Poisson integrals and \(\psi_1\)-chaining. Despite the weaker concentration, the resulting statistical bound is, up to constants, of the same order as the corresponding diffusion bound. For drift functions with hierarchical compositional structure, this yields intrinsic-dimensional convergence rates. The framework allows infinite jump activity and, in some cases, infinite variation. Finally, we establish a minimax lower bound on a fixed non-trivial compound-Poisson submodel. Together with the upper bound, this shows that the estimator is minimax optimal up to logarithmic factors.

We consider the interaction of a viscous incompressible fluid with a flexible elastic shell occupying the boundary of the fluid's domain in two space dimensions. The shell is modelled by a perfectly elastic linearised beam equation. It is an open problem in fluid-structure interaction, even for the idealised case of a perfectly elastic plate, if a self-intersection of the structure can be excluded in the long run. We show that by adding a physically well-motived and sufficiently strong transport noise in the structure equation, the self-intersection does not occur before any given end-time with any prescribed probability smaller than 1.

For a positive real lacunary sequence and any probability measure $\mu$ with polynomial Fourier decay, we determine, for $\mu$-almost every $x$, the asymptotic maximal volume of convex bodies avoiding the first $N$ dilates of $x$ modulo $\mathbb Z^d$. Among homothets of a fixed convex body, the maximal volume is asymptotic to $(\log N)/N$. When all convex bodies in the unit cube are allowed, it is asymptotic to $d(\log N)/N$. For independent uniform points in a fixed convex body $\Omega\subset\mathbb R^d$, the maximal volume of an empty convex body is asymptotic to $d\operatorname{vol}(\Omega)(\log N)/N$ almost surely. We also show, by constructing a one-dimensional counterexample, that lacunarity cannot be replaced by sparsity on sublacunary scales.

Mourad E.H. Ismail and Kenneth S. Miller in 1982 conjectured that for $\nu>\mu\geq0$ and $b>a>0$ the function $$x\mapsto G(x;\mu,\nu)=\left(\frac ba\right)^{\mu-\nu}\frac{I_\mu(a\sqrt{x})I_\nu(b\sqrt{x})}{I_\mu(b\sqrt{x})I_\nu(a\sqrt{x})},$$ where $I_\nu$ stands for the modified Bessel function of the first kind, is the Laplace transform of an infinitely divisible probability distribution. Their conjecture is equivalent to the monotonicity property of some exponential sums over the zeros $j_{\nu,n}$ with respect to the order, where $j_{\nu,n}$ denotes the $n$th positive zero of the Bessel function of the first kind of order $\nu$. In this paper our aim is to reduce the problem further to the monotonicity, with respect to $\nu$, of \[Q_\nu(t)=\sum_{n\geq1}j_{\nu,n}^2e^{-t j_{\nu,n}^2},\qquad t>0,\] and to prove this monotonicity. In this way we show that indeed the above function $x\mapsto G(x;\mu,\nu)$ is the Laplace transform of an infinitely divisible probability distribution, and the above conjecture is true. The key ingredient in our proof is an integral representation for the derivative of a quotient of modified Bessel functions of the first kind with respect to the order, combined with Bernstein's theorem and a probability tail argument.

Mean field limits describe the training dynamics of wide neural networks through the evolution of the empirical distribution of their parameters. Although functional central limit theorems characterize the asymptotic fluctuations of this distribution, quantities of practical interest are typically nonlinear observables of the parameter distribution rather than the distribution itself. In this work, we show how these mean field fluctuations propagate to finite dimensional nonlinear observables for shallow neural networks trained by stochastic gradient descent. Working in the weighted Sobolev space in which the limiting fluctuation process is constructed, we apply a functional Delta method under ordinary Fréchet differentiability, without requiring Lions derivatives with respect to the measure variable. We obtain a central limit theorem for the observables and, under a suitable representation of their differentials, an explicit covariance formula inherited from the underlying mean field fluctuation theory. We also study whether prescribed quantities of interest can be recovered from the selected observations. Under a constant rank assumption, we prove that a quantity of interest factors locally through the observation functional if and only if, throughout a neighborhood, the kernel of the differential of the observation is contained in that of the quantity of interest. Thus, a differential condition expressed directly in the ambient Sobolev space yields an exact nonlinear local factorization. These results provide a framework both for quantifying finite-width uncertainty on observable, statistically or physically meaningful quantities and for assessing whether the chosen observations contain the information required to identify them.

We study the gap between the integrated direction-wise kinetic minima and the minimum expected Dirichlet energy of a joint particle realization of a prescribed two-parameter family of conditional probability measures. Under regular coarea assumptions with compact connected fibres and a star-shaped parameter domain, we derive a curvature lower bound valid for every admissible law of Sobolev maps realizing the prescribed smooth positive conditional measures. Our method combines the weighted-Poisson characterization of local optimal-transport velocities with an exact energy-residual identity, a weak Stokes formula, and explicit transport constructions. The minimum excess vanishes exactly when the Lie-bracket curvature vanishes throughout the parameter domain, whereas at a nonzero-curvature centre it is of order $r^4$ as $r\to0$ on squares of half-side length $r$ with unnormalized area measure. In a two-phase torus model fitted to electricity-demand data, numerical tests illustrate this quartic order for explicit couplings, with radial transport approximately halving the ordered construction's excess on the tested small squares.

Let $f_n:\{0,1\}^n\to\{0,1\}$ be Boolean functions, and let $C_n$ count their changes in a unit time interval under stationary dynamics in which each coordinate is independently resampled at rate one from the Bernoulli$(p_n)$ distribution. We prove that, if $f_n$ is transitive and $n\min\{p_n,1-p_n\}^{r}\to\infty$ for every $r>0$, then tightness of $(C_n)$ implies $\mathrm{Var}(f_n)\to0$. This settles Forsström's conjecture in the $p_n\le 1/2$ regime and extends it symmetrically to arbitrary biases. The proof uses a mixed jump sum whose second moment is bounded by the generator of a low degree Fourier projection.

We present a novel dual approach to characterizing the solution to optimal stopping problems for strong Feller processes on locally compact, separable metric spaces. Our characterization builds upon reinterpreting a standard result in optimal stopping theory as a dual formulation, which yields computationally tractable algorithms. Our framework is applicable to a broad class of Markov processes. First, in the one-dimensional case, while previous studies have managed to comprehensively solve only diffusion and one-sided jump Lévy processes over an infinite time horizon, our proposed method successfully extends to both infinite- and finite-horizon scenarios across all these processes, including two-sided jump Lévy processes. In addition, for multi-dimensional cases, our framework is applicable to diffusions over an infinite horizon with certain structural features such as convex stopping regions. To illustrate this versatility, we explicitly solve several challenging problems including a case where the optimal stopping region consists of two disjoint connected components – a structural feature that remains unaddressed in the literature.

We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an $L^p$ density, $p>1$, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the $W_2$ Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the $L^p$ norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For $L^p$ changes of Gaussian measure, the factor in the bounded-target estimate is $d^{(p-1)/(4p-2)}$. For each $p$, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.

Let an Itô semimartingale be observed at times satisfying $\tau_{i+1}^n-\tau_i^n=(n\theta_{\tau_i^n})^{-1}$, where $\theta$ is adapted and càdlàg, and $\theta$ and its reciprocal are bounded on compact intervals. Under the usual hypothesis (H) on the semimartingale, we prove fixed-time stable convergence for test functions whose Hessian is $o(\norm{x})$ at zero. The limit is a sum of jump contributions with sampling scale $\theta_{T-}^{-1}$ and separate pre-jump and post-jump volatilities. The proof establishes joint stable convergence of the local Poisson positions and Brownian increments on the endogenous grid, and controls infinitely many small jumps by a uniform Itô estimate. Three examples identify the left-limit sampling scale and show why positivity and càdlàg regularity alone cannot replace local control of the reciprocal sampling intensity.

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

We study a strongly damped stochastic wave equation on the three-dimensional torus driven by finite-rank pure-jump Lévy noise obtained from subordinated Brownian motion. Lyapunov estimates and an asymptotic compactness decomposition yield invariant measures, and a small-noise event gives weak irreducibility. To prove uniqueness, we combine conditional Malliavin calculus with terminal observability and estimates for the full variational flow starting from high modes. A control constructed on alternating stopping blocks yields the e-property under a fixed-rank balance condition involving high-mode decay, spectral leakage, and the random clock. Consequently, the invariant measure is unique and ergodic, and the Cesàro averages of the transition laws from every initial state converge weakly to it. We verify this condition for sufficiently small positive cubic coefficients at each fixed forcing operator of sufficiently large rank. Under a separate explicit smallness condition, synchronous coupling proves the e-property and weak convergence of the transition laws, without a lower bound on the forcing rank.