Papers

We prove a dimension-free Markov–Bernstein-type inequality for polynomial mappings $(f_1,f_2)\colon\mathbb R^n\to\mathbb R^2$ of degree at most $d$. For product probability measures whose one-dimensional marginals have densities bounded by $1$, we obtain a lower bound for the integral of the Gram determinant of $\nabla f_1$ and $\nabla f_2$ in terms of the Gram determinant of the $d$-th derivative tensors $D^df_1$ and $D^df_2$, with a constant depending only on $d$. For pairs in a fixed Wiener chaos, the resulting Gaussian inequality answers a question of Nourdin, Nualart, and Poly concerning a lower bound for the expectation of the Malliavin determinant in terms of the determinant of the covariance matrix.

We study the Euler-Maruyama approximation of the degenerate stochastic differential equation $dX_t = |X_t|^\alpha\,dW_t$, $0 < \alpha < 1/2$, $X_0 = 0$, for which uniqueness in law fails. The Euler scheme started exactly from zero is trapped by the absorbing solution, whereas a nonzero initialization $x_h \to 0$ can select a different weak solution. We identify the initialization-dependent limiting regimes. The critical spatial scale is $r_h = h^{1/[2(1-\alpha)]}$, and the relevant depth parameter is $\Theta_h = h\log(1+\log^+(r_h/|x_h|))$. If $\Theta_h \to 0$, the scheme converges to the non-sticky weak solution. If $\Theta_h \to \theta \in (0,\infty)$, it converges to the solution that stays at zero for the deterministic time $\theta/|\log\alpha|$ and then evolves as the non-sticky solution. If $\Theta_h \to \infty$, the scheme converges to the absorbing solution. The proof is based on an exact scale-invariant Euler chain, a logarithmic exit-time law of large numbers, a discrete Tanaka occupation estimate, and a compact-uniform restart argument. Every finite deterministic delay can be obtained by a suitable vanishing initialization, while oscillating depth parameters yield process-level nonconvergence. Thus a vanishing initialization can survive in the limit as a macroscopic deterministic delay and thereby act as a solution selector for the Euler-Maruyama approximation.

We construct a linear map from the space of spin polynomials in the critical Ising model on a lattice to the state space of the $M(4,3)$ minimal model of conformal field theory. The kernel of the map consists precisely of null fields, i.e., spin polynomials that vanish in all correlations. For a polynomial $\Op$ which is not a null field, we define its scaling dimension $\Delta_{\Op}\in\Z_{\geq0}\cup(\Z_{\geq0}+\frac{1}{8}),$ to be the lowest $\Loper{0}+\Loperbar{0}$ eigenvalue in the decomposition of its image into homogeneous components. We show that as the mesh size $\delta$ goes to zero, the rescaled field $\delta^{-\Delta_{\Op}}\Op$ has a non-trivial scaling limit, when inserted in correlations with other such fields. Moreover, all these limits can be expressed in terms of correlations of primary fields of $M(4,3)$ and the Virasoro action in the continuum. The highest weight vectors of $M(4,3)$, the joint eigenvectors of $\Loper{0}$ and $\Loperbar{0}$ with the same eigenvalues $0,1/16,1/2$ for both, are the images of the constant $1$, the spin at a single lattice vertex, and the energy at a single lattice edge, respectively. For the proof, we construct a map from a larger lattice field space, accommodating also fermion and disorder insertions, to a direct sum of two fermionic Fock spaces. This direct sum contains contains the state spaces of both the $M(4,3)$ and the free fermion CFT. The main technical ingredient of the proof is a new version of the exact fermionic current action on the space of lattice local fields due to C. Hongler, K.K. and F. Viklund, combined with some new advances in discrete complex analysis. Our results are valid on arbitrary isoradial graphs.

We consider the Burgers-type stochastic PDE \[ du = \theta\,\partial_x^2 u\,dt + f(u)\,dt + g(u)\,\partial_x u\,dt + \sigma\,dW(t), \] driven by space–time white noise on the one-dimensional torus, where \(\theta>0\) is the diffusivity parameter and \(\sigma>0\) the noise intensity. The transport term \(g(u)\,\partial_x u\) admits no classical interpretation, and we work with the mild solutions defined via rough paths following~\cite{Hairer2011}, for which the solution decomposes as \(u=\psi+R\) with \(\psi\) a stationary centred Gaussian process solving a damped stochastic heat equation and \(R\) a remainder. We first show that, away from the initial time, \(R\) is Lipschitz in space and Hölder in time of order strictly greater than \(1/2\), hence strictly smoother than \(\psi\) in both variables. We then compute exactly the spatial quadratic variation and the temporal quartic variation of \(\psi\), almost surely and not merely in \(L^{2}(\Omega)\), and prove that the corresponding variations of \(u\) are unaffected by the remainder. Combined with an exact time-scaling identity, this separation of regularity scales yields strongly consistent pathwise estimators of \(\theta\). To our knowledge this is the first inference procedure for an SPDE whose solution is defined only pathwise, as a measurable function of a rough path, rather than as an Itô process.

We establish a Thouless–Anderson–Palmer (TAP) variational formula for the free energy of spherical $2$-spin models whose interaction matrices have a compactly supported limiting spectral distribution. The formula holds at every inverse temperature and, in particular, extends the TAP representation previously obtained for the spherical Sherrington–Kirkpatrick model. Our main contribution is a new microcanonical proof designed to handle marginality. We partition the sphere according to the smallest collection of top eigendirections carrying a prescribed fraction of the norm. On every element of this partition, the remaining degrees of freedom are effectively at high temperature and their contribution can be computed through spherical-integral asymptotics. The resulting auxiliary variational problem separates the macroscopic low-temperature contribution from the high-temperature fluctuation term; its maximization recovers the usual TAP functional and Plefka condition. This decomposition also shows why neighborhoods of the TAP maximizers alone do not capture the Gibbs mass: all eigendirections associated with large eigenvalues contribute on the same exponential scale, a manifestation of the marginality of spherical $2$-spin states.

We study sampling of a diffusion on $[0,T]$ conditioned on its terminal position lying on a prescribed ray, i.e. on an exactly observed bearing. For a linear drift we derive closed-form expressions for the Doob $h$-function and its gradient. For a broad class of tracking models admitting a Kronecker block structure, the guiding term is obtained from a single row of a matrix exponential, and the conditioned process decomposes into independent radial and angular components. For a nonlinear drift, we construct a guided process using the $h$-function of the linear model and prove absolute continuity of the law of the true conditioned process with respect to the law of the guided process, with an explicit likelihood ratio. This result is used to devise an auxiliary particle filter for bearings observed sequentially in time without noise. Numerical experiments show that the proposed filter approximates the filtering distribution and estimates the log-likelihood considerably more accurately than the shifted Rayleigh filter, at the cost of a higher computing time.

We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on $\mathbb{R}^d$ with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time $T$ is bounded by $C(1+T)e^{-cN}$, where $N$ is the number of particles and $C,c>0$ are independent of $N$ and $T$. Consequently, confinement holds up to times $e^{\gamma N}$, for every $0<\gamma<c$, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.

We establish the following novel heterovariant Gaussian Brunn-Minkowski inequality (involving general covariance matrices $\Sigma_1,\Sigma_2,\Gamma$) for origin-symmetric convex sets $K,L \subset \mathbb{R}^n$ and $p,\lambda \in (0,1)$: \[ \gamma_{\Sigma_1}(K)^{1-p} \gamma_{\Sigma_2}(L)^p \le \left( \left( \frac{1-p}{1-\lambda} \right)^{1-p} \left( \frac{p}{\lambda} \right)^p \right)^{n} \gamma_{\Gamma}((1-\lambda) K + \lambda L) , \] whenever $\Gamma \leq \Sigma_1 \sharp_p \Sigma_2$, the $p$-geometric mean of $\Sigma_1,\Sigma_2$. This immediately implies both the dimensional Gaussian Brunn-Minkowski inequality of Eskenazis-Moschidis (by using $\Sigma_1 = \Sigma_2 = \Gamma$ and optimizing over $p$) and the strong Gaussian (B)-theorem of Cordero-Erausquin-Fradelizi-Maurey (by using $K=L$, $p=\lambda$ and diagonal covariances), fusing them into a single unified statement. In fact, the inequality applies to convex sets whose appropriate Gaussian barycenter is at the origin. Furthermore, it reveals a new *bilinear* version of the strong Gaussian (B)-theorem (involving *two* convex sets); in particular, for all $a,b > 0$ and $\lambda \in [0,1]$: \[ \gamma(a K)^{1-\lambda} \gamma(b L)^{\lambda} \leq \gamma \left ( a^{1-\lambda} b^\lambda \left ((1-\lambda) K + \lambda L \right) \right ) . \] A more general heterovariant version of the (bilinear) (B)-theorem replaces dilation by general linear transformations. A different version yields a heterovariant form of the recently established Gaussian conjugate Rogers-Shephard inequality. By averaging these Gaussian inequalities with respect to the covariance matrices, we obtain dimensional Brunn-Minkowski and conjugate Rogers-Shephard inequalities for Gaussian mixtures. The proofs are based on the Gaussian Forward-Reverse Brascamp-Lieb inequality established in our previous work.

[Bass and Burdzy 1999] introduced a stochastic flow of homeomorphisms of the real line $\mathcal R_t:x\mapsto \mathcal R_t(x)$, driven by a Brownian motion. Ray-Knight theorems for the process $\mathcal R_t^{-1}(0)$ were later obtained by [Hu and Warren 2000]. We extend their results by proving an integral representation driven by a common white noise for the local times of $\mathcal R_t^{-1}(0)$, in analogy with the case of the standard Ray-Knight theorems [Aïdékon, Hu and Shi 2024]. We generalize this representation to the setting of a Bass-Burdzy flow driven by a drifted Brownian motion perturbed at its maximum, and use this framework to answer a question of [Aïdékon, Hu and Shi 2026] concerning the description of the conditional law of a perturbed reflecting Brownian motion given its occupation measure in terms of Bass-Burdzy flows.

We study a stationary ergodic mean-field game of scalar type where the representative player controls a diffusion through two-sided singular controls exerted only at the jump times of an independent Poisson process. For each mean-field parameter, we solve the novel constrained ergodic singular control problem, proving that the optimal policy is of threshold type: the diffusion is reset at Poisson arrival times to the nearest boundary point of an interval. Our explicit probabilistic derivation of the stationary distributions induced by Poissonian threshold singular controls is a key contribution of independent interest. This distribution is used to construct the mean-field fixed-point map. We prove existence and uniqueness of the stationary mean-field equilibrium by combining the stationary density with the monotonicity and continuity of the optimal thresholds in the mean-field parameter. The model provides a tractable framework for a case study of partially reversible investment with singular adjustments under exogenous intervention constraints.

We study the zeros of the spectrogram of complex white noise analyzed with Hermite windows of order n, extending the classical Gaussian-window correspondence with Gaussian Analytic Functions to the polyanalytic setting. Using the polyanalytic Bargmann transform, these zeros split into two families according to the sign of the local Jacobian, giving rise to a natural notion of charge, and we show that these zeros are almost surely simple, isolated, and form a translation and rotation-invariant point process. We derive closed-form expressions for the one-point intensity, decomposed by charge, and also for the unsigned, signed, and hybrid two-point intensities. Besides, we prove that unlike the total repulsion observed with a Gaussian window, the Hermite window of order n generates a pair correlation function that converges at the origin to a strictly positive constant depending on n. Finally, we propose a numerical method for detecting spectrogram zeros via the winding number around candidate local minima.

In this paper, we investigate the weak invariance principle in some Besov spaces for strictly stationary sequences. In particular, we provide sufficient conditions on the moments and the dependence of a strictly stationary sequence to guaranty the weak invariance principle in these Besov spaces. Several dependence structures are considered, including projective conditions, $\alpha$-dependent, $\rho$-mixing, and $\tau$ -dependent conditions. The main tools in the proofs are martingale approximation and deviation inequalities for weakly dependent sequences. Our results cover a large range of strictly stationary sequences that were not treated previously in the literature. Moreover, the presented conditions are sharp.

Semi-Markov networks provide a natural framework for describing stochastic transitions between coarse-grained states with non-exponential dwell times, with applications ranging from biophysics and neuroscience to epidemiology and finance. Here, it is shown that finite semi-Markov networks admit explicit combinatorial representations of their Laplace-transformed occupation probabilities and hitting-time distributions as sums over weighted spanning trees and forests of an augmented graph. These representations yield asymptotic results, including a generalization of the Markov Chain Tree Theorem to semi-Markov networks, in which the rate constants replaced by pseudo-rates given by the inverse mean sojourn time weighted by the exit probability through this channel, as well as the long-time scaling of hitting-time survival probabilities in the presence of infinite-mean sojourn times. They also extend recently derived combinatorial formulas for hitting-time moments to the semi-Markov setting. In particular, the mean hitting-time formula coincides with that of a Markov network with the corresponding pseudo-rates, which also yields a short proof of the constancy of Kemeny's constant for finite-mean semi-Markov networks. Finally, we use the same framework to show that weak ergodicity breaking occurs generically for time-additive state observables that are nonconstant on the support of the limiting distribution in heavy-tailed semi-Markov networks.

We prove the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the large-distance behaviour of the two-point correlation function of the $\mathsf{Sine}_\beta$ point process for all $\beta>0$, confirming the first term of an asymptotic expansion predicted by Forrester.

Recovery risk measures incorporate explicit liability-recovery requirements into asset-liability risk assessment. We study their specialization to optimized certainty equivalents (OCEs), allowing the underlying utility specification to vary across recovery levels and encompassing Recovery Average Value at Risk (RecAV@R) as a special case. For finite recovery specifications, we use the dual penalty representation to characterize redundant recovery regimes and derive position-independent screening conditions. For a subfamily of the Cressie-Read power-divergence family, including entropic and truncated-quadratic specifications, we obtain an exact characterization on atomless probability spaces: universal redundancy is equivalent to a two-dimensional convex-hull condition involving recovery levels and divergence-scale parameters. On such spaces, universal redundancy for RecAV@R reduces to pairwise dominance, whereas for power-divergence OCEs it can arise from the joint effect of several recovery regimes. For nonlinear OCE utilities, we develop a certified piecewise-linear approximation framework for portfolio optimization. In finite-scenario models, the approximating problems are finite-dimensional linear programs, while utility approximation errors transfer directly to the Recovery OCE, yielding explicit error bounds and convergence rates. Uniform approximation over the portfolio domain further implies convergence of optimal values and stability of optimal solutions. A numerical asset-liability study illustrates the effects of recovery-dependent risk preferences and the accuracy and convergence of the approximation.

We study the expectation of the discrete total variation statistic $d=\sum_{i=1}^{n-1}|X_{i+1}-X_i|$ of counts $X_1,\dots,X_n$ over $n$ ordered bins with sample size $N$, under the uniform multinomial null. In the notation used throughout the series, a reference distribution $\mathbf r$ defines $d_{\mathbf r}(x)=\|x-N\mathbf r\|_V$; here $\mathbf r=\mathbf u$ is uniform and we write simply $d$. We obtain an exact single-sum formula, a closed form in central trinomial coefficients valid for every $n\ge2$, with one-term evaluations at $n=2,4$ and a complete D-finite characterization for $n=3$. The per-edge expectation $W_N=E|X_2-X_1|$ satisfies an exact three-term recurrence, from which we derive its complete $1/N$ expansion with the $n$-independent constant $E(d^\circ/\sqrt{Nn})\to 2/\sqrt\pi$ for the cyclic statistic $d^\circ=d^\circ_{\mathbf u}$, independent of the number of bins; the complementary sparse regime (fixed $N$, $n\to\infty$) is treated in a separate paper. Conditioning on a two-bin total reduces each edge to a symmetric random walk; as the subsequent parts of the series will show, this method carries the analysis to non-uniform reference laws.

We prove finite-sample risk bounds for empirical quadratic and power Fréchet means in geodesic metric spaces with curvature bounded above for observations with arbitrary joint law. We compare the sample with exchangeable copies in which one observation is redrawn independently of the original sample and the others move to preserve the joint law. These displacements control the error, with no entropy or dimension assumptions. For quadratic Fréchet means, this yields a general “coupling bound” in any separable complete $CAT(\kappa)$ space, where for $\kappa>0$ the observations must lie in a closed convex set of circumradius below $\pi/(2\sqrt{\kappa})$. This bound is exact on the real line for nonnegatively correlated Gaussian observations and for comonotone functions of a common factor. We apply it in two settings. Under local dependence, the resulting risk bound grows linearly with the average size of the dependency neighborhoods. Under transport sensitivity and for bounded metric variances, summable sensitivities of the conditional laws to a change in one observation preserve the parametric mean squared error rate, while nonsummable polynomial decay gives explicit slower rates. We verify transport sensitivity for stationary reversible Markov chains with Wasserstein-contractive kernels and for bi-Lipschitz images of Gaussian sequences. In the more challenging setting of heavy-tailed, dependent data with possibly infinite variance, we consider power Fréchet means of order $\alpha\in(1,2)$. Under nonpositive curvature, we bound their $L^\alpha$ risk assuming only finite $\alpha$-th moments.

Quantitative Gaussian approximation estimates on path space are established for the nonequilibrium density fluctuations of the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus of mesh size $1/n$, started from a product measure with a smooth density profile. A novel $n$-dependent modified Ornstein-Uhlenbeck process is constructed that incorporates higher-order lattice information while remaining linear and Gaussian. It is shown that, for sufficiently high modification orders, this modification yields quantitative Gaussian approximation error bounds of order $n^{-\min\{d/2,2\}}$ for a class of smooth functionals of the entire fluctuation trajectory. In addition, it is shown that retaining an explicit microscopic correlation correction leaves a remainder of order $n^{-d/2}$, thereby distinguishing contributions to the bound due to correlations from those due to spatial approximation. For nonconstant initial profiles in dimensions $d\ge3$, a separate argument yields matching upper and lower bounds of order $n^{-2}$ for the direct weak error of suitable quadratic test functionals, provided that the modification orders are sufficiently high. The proof is based on a novel path-dependent Trotter formula. Its derivation builds on the introduction of a concept of horizontal absolute continuity for non-anticipative functionals and a corresponding martingale formula for Markov jump processes. This formula is subsequently combined with an integral backward Kolmogorov equation for Gaussian continuation functionals. A second ingredient of the proof is a refinement of the exchangeable-pair argument introduced by the authors in a previous work, thereby improving the central-limit rate $n^{-\min\{d/2,1\}}$ to $n^{-d/2}$ for the initial empirical fluctuations.

Motivated by semi-Markov models of anomalous diffusion, we study obliquely reflected diffusions in generalized parabolic domains under independent inverse-subordinator time changes. We focus on infinite-mean subordinators, whose inverse clocks can slow the superdiffusive growth induced by reflection. For domains with cross-sectional radius asymptotic to $a_\infty x^\beta$, where $-1<\beta<1$, we strengthen the almost-sure growth laws of [MMW23] to obtain precise second-moment asymptotics when the subordinator's Laplace exponent is regularly varying at zero, and matching bounds more generally. Inverse $\alpha$-stable clocks yield mean-square growth of order $t^{2\alpha/(1+\beta)}$, giving subdiffusive, diffusive, or superdiffusive regimes. For $\beta<-1$, explosion persists almost surely, with finite mean explosion time if and only if the subordinator has finite mean. Supporting results establish strong existence, pathwise uniqueness, and a time-change representation, also allowing dependent clocks.

We consider the Navier-Stokes equations in a smooth bounded domain $D\subset \mathbb R^3$. It has been proved in arXiv:2609.08284 that these equations have a unique solution - globally defined in time - when a superlinear noise is introduced and the initial velocity belongs to the space $V$ of divergence-free $H^1_0(D;\mathbb R^3)$-vector fields. Here we show that the same result holds when there is also an additive noise and a deterministic forcing term. Moreover, when the deterministic forcing term is time-independent we prove the existence of at least one invariant measure for this stochastic equation. This is obtained by the Krylov-Bogoliubov method, based on tightness and Feller property in $V$. Finally, we show that the set of invariant measures contains one ergodic invariant measure.

For diffusions whose coefficients are smooth, with at most linear growth and bounded derivatives of higher order, we establish explicit pointwise bounds on the spatial derivatives of the classical solution to the Cauchy problem with polynomially growing initial data. We also obtain explicit growth estimates for the derivatives of the SDE with respect to the initial data. The proofs combine Lyapunov functions and Markov semigroups on weighted spaces with Kunita's theory of the differentiable dependence of SDE solutions on the initial data.

We study penalizations of the two-dimensional simple random walk by functions of direction-dependent excursion counts and edge-crossing counts. The former record completed excursions with prescribed pairs of departure and return directions at the origin, whereas the latter record crossings of a prescribed set of unoriented edges incident to the origin. We determine the distributions of their almost surely finite terminal values and derive explicit path decomposition formulas involving the random walk conditioned to avoid the origin. Moreover, both families of penalized measures can be represented by weighting and normalizing the same universal $\sigma$-finite measure associated with local-time penalization at the origin. Thus, direction-dependent excursion counts, edge-crossing counts, and local time at the origin give rise to penalizations belonging to a common universality class.

Random spanning forests provide probability models for partitions and Monte Carlo methods on graphs. In these applications, the distribution produced by a sampler determines which forest functionals can be estimated. The Aldous–Broder algorithm constructs a spanning tree from the first-entrance edges of a covering random walk. Its distribution is known, but a direct calculation requires summing over all orders in which a fixed tree can be discovered. We evaluate this sum through an identity for rooted forests and real matrices with zero row sums. Under a nonsingularity condition on the principal submatrices encountered during growth, the sum of products of harmonic extension entries equals a principal cofactor of a matrix on the roots. With one root, the sum is one. The identity gives a direct proof of the extended Aldous–Broder theorem for every finite irreducible Markov chain started at a fixed state, without assuming reversibility or bidirectional transition support. It also gives the conditional distribution of the discovery order and the law of the forest obtained by stopping after all nonroot vertices have been visited. A completion weight describes the difference between this stopped-walk law and the weight-proportional forest law, including the support restriction needed for importance correction. Numerical examples illustrate the sampling implications. The identity, the Aldous–Broder theorem, and the stopped-forest law are formalized in Lean4.

After centering and scaling, the length of the longest weakly increasing subsequence of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting RSK Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the limiting Brownian functional may fail to have the law of the largest eigenvalue of any centered jointly Gaussian $3\times 3$ Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.

We study quantum-assisted balanced rendezvous on the $d$-dimensional hypercube in a one-step two-agent quantum-walk framework, where the two agents move the same Hamming distance and meet at a common midpoint. The key structure is a projective $t$-sphere resolution, which governs exact finite-dimensional quantum attainment. Its scalar analogue recovers the corresponding classical characterization. We characterize finite-dimensional quantum attainment of the no-signalling benchmark, construct quantum strategies from projective sphere resolutions, and establish a rigidity theorem for optimal finite-dimensional strategies. A marked difference appears between nearest-neighbor and higher-cardinality updates: quantum advantage can occur in the former, whereas in the latter the geometry of the relevant Hamming spheres rules out exact finite-dimensional attainment of the no-signalling value. Together, these results connect quantum-walk dynamics, Hamming-sphere geometry, and quantum rendezvous within the quantum Bernoulli noises (QBN) framework.

Permutons can be thought of as "continuum limits" of permutations of diverging size, which encode information about the number of occurrences of any given finite subpattern. We construct a natural composition operation on permutons, analogous to composition of permutations, and explore its properties via a correspondence between permutons, Markov kernels, and Markov operators. We show that no extension of permutation composition to permutons can be (jointly) continuous with respect to the usual "weak" topology on permutons, but we construct a "strong" topology with respect to which composition is jointly continuous. Moreover, we characterize strong convergence of sequences of permutations of diverging size: a sequence of permutations converges strongly if and only if it converges weakly and its limit is the transpose of the graph of a measure-preserving function. The key insight behind these results is to apply functional analysis to certain operators associated to permutons; with this shift in perspective, our results have elegant and relatively "soft" proofs. As an application, we describe the permuton limit of a product of finitely many independent uniformly sampled separable permutations and give a formula for the expectation of this limit. We also show that in this case, as the number of compositions goes to infinity, the aforementioned expectation tends to the uniform permuton.

We study the asymptotic properties of the nonlinear stochastic GMM algorithm proposed by Chen, Lee, Liao, Seo, Shin and Song for recursively estimating parameters defined by possibly nonlinear and overidentified moment restrictions $\E[g(X,\beta^*)]=0$. Each observation updates estimates of the moment Jacobian, inverse covariance matrix and Gauss–Newton preconditioner at a cost independent of sample size. For linear and nonlinear instrumental-variable moments, we derive an exact rank-two Sherman–Morrison–Woodbury update. The algorithm combines preconditioned stochastic approximation with iterate averaging. Under a preconditioned monotonicity condition, satisfied globally in linear models and locally around a consistent pilot estimate in general, we establish almost sure convergence and rates, as well as a functional central limit theorem with the efficient GMM variance. These results require four moments of the moment function and its Jacobian when the step exponent exceeds $2/3$. The inverse Gauss–Newton matrix consistently estimates this variance, and a projected linearised statistic enables online testing of overidentifying restrictions. We extend the results to $L$-dependent data with long-run variance weights under stronger moment assumptions. Simulations show that the averaged estimator approaches offline two-step GMM in accuracy and coverage at $10^6$ observations, while requiring a fraction of its computing time and fixed memory at up to $10^7$ observations. The online test achieves correct size and comparable power. In an application estimating the effect of a third child on weeks worked using $254{,}654$ women from the 1980 US Census, single-pass estimates agree with offline GMM within a fraction of a standard error on average across data orderings, and both methods yield the same overidentification-test conclusion in most orderings.

We study Gaussian maxima arising from isotropic families of unit vectors and show that their excess expected maximum is controlled by the distance from an orthonormal basis. We apply this stability result to correlation matrices with equal nonzero eigenvalues, obtaining an exact minimum when the matrix size is divisible by the rank and a quantitative gap otherwise. In dimension two, we determine the leading asymptotic term and describe the structure of near-optimal configurations. We also obtain covariance-block recovery results and a finite-dimensional lower bound for the higher-rank limiting problem.

We study a model of a random centrally symmetric polytope and ask about the existence of points in the polytope with non-zero, integer coordinates. Our main results are the location of a sharp threshold for the existence of these integer points, a characterization of the typical empirical distribution of their coordinates, and showing that near the feasibility threshold these integer points are almost always totally isolated — at linear Hamming distance to the nearest other integer point in the polytope. The behavior of these random integer programs is related to that of the symmetric binary perceptron (the case when coordinates of points are $\pm 1$-valued). The lack of symmetry in the set of potential solutions leads to new questions and phenomena, such as the distribution of the coordinate sizes in typical solutions, and requires the development of new information-theoretic techniques to deal with higher-dimensional optimization problems arising in the method of moments.

We prove a large deviation principle for additive functionals of local Lipschitz functions of a one-dimensional symmetric zero-range process on the diffusive scale. The rate function is obtained from the dynamical large deviation principle for the empirical density and is characterized by a variational constraint involving the density at a fixed spatial point. On the microscopic side, a super-exponential replacement is needed to replace the local observable with a function of the empirical density. On the macroscopic side, the spatial regularity enjoyed by finite-rate trajectories is needed to pass from spatial averages to point evaluation.