Papers

This paper establishes the existence of local martingale solutions for a three-dimensional stochastic fluid–structure interaction (FSI) problem coupling a viscous compressible isentropic fluid with an elastic shell through Navier-slip boundary conditions. The fluid and shell are perturbed by compatible Stratonovich transport noise. A stochastic-flow transformation removes the stochastic transport integrals and produces a pathwise random FSI system whose pullback coefficients are only Hölder continuous in time. We combine a multi-layer approximation scheme with localization on bounded-flow events and freezing of the random coefficients on short deterministic time subintervals. The shell transport fields are not required to generate isometries: the bending-energy martingale and Itô correction are controlled at the natural $H^2$ level by commutator identities without derivative loss. Tightness is obtained in Jakubowski spaces, pressure compactness is proved through localized effective viscous flux, and the final artificial-pressure and penalty limit uses boundary-pressure tightness and $H^2$-graph trace tools. For $\gamma>3/2$ and non-colliding initial data, we obtain a martingale solution up to an almost surely positive stopping time.

Consider an $N\times N$ random band matrix $H$ with bandwidth $W$ with centered real Gaussian entries. We prove that, under the (almost) sharp assumption $W^2\ll N/\log N$, the bulk local statistics of $H$ are asymptotically a Poisson point process. Combined with the bulk universality results from arXiv:2501.01718 and arXiv:2506.06441, this establishes the Poisson-RMT transition of the local statistics of random band matrices, and complements the results about the localization-delocalization transition of the bulk eigenvectors previously established in arXiv:2501.01718 and arXiv:2506.06441, and arXiv:2508.05802. The key technical inputs are: (1) the fractional moment estimates from arXiv:2508.05802; (2) a novel estimate for the density of states, obtained from the local laws in arXiv:2501.01718 and arXiv:2506.06441.

We show that full negative regression dependence implies neither negative right-tail dependence nor negative left-tail dependence. A strictly positive distribution on a 2*2*3 rectangular state space provides an exact counterexample for the right-tail case, and its coordinatewise reflection yields a counterexample for the left-tail case. We further prove that this state-space size is minimal. The counterexample is established via a finite upper-set criterion using exact integer arithmetic. We also resolve the open question of whether full negative regression dependence implies negative association. A strictly positive distribution on {0,1,2}^4 satisfies full negative regression dependence yet violates negative association; this counterexample is verified by 17532 exact integer inequalities. The main results of this paper were first uncovered by the multi-agent mathematical system Eureka and later verified by the authors.

In this article, we establish central limit theorems for sample Fréchet means of stationary ergodic Markov chains taking values in manifolds, extending the asymptotic theory previously developed for independent observations to a class of dependent manifold-valued processes. Our results derive the asymptotic normality of the sample Fréchet mean from a central limit condition at the population Fréchet mean, under suitable local regularity conditions. We further provide sufficient geometric and probabilistic conditions under which these assumptions hold, formulated in terms of curvature bounds and a Wasserstein mixing condition. As an application, we establish a central limit theorem for sample Fréchet means for a class of random dynamical systems generated by contractive random maps.

We construct finite-state Markov arrival-rate processes for which faster modulation increases the mean stationary workload of a stable single-server queue, disproving the universal ordering in Leskelä's formulation of Ross's second conjecture. A 64-state example has positive transition rates between every pair of states, independent service times uniform on $[0.99,1.01]$, and an explicit positive workload gap. A 128-state example additionally has strictly positive arrival-rate autocovariance at every lag. More generally, finite-state examples can exhibit any prescribed finite number of separated workload rises and falls. The proofs use a second-order expansion with a nonnegative, speed-uniform remainder under only a finite second service moment. We characterize the service distributions for which the second-order workload coefficient is nonincreasing for every bounded stationary environment: the function $s\mapsto\mathbb{E}(1-\cos(sB))/s^2$ must be nonincreasing. Mixtures of exponential service distributions satisfy this criterion. An exact Palm identity transfers the workload reversals to customer waiting times.

Self-calibrating monitors adapt their threshold online to guarantee a prescribed long-run false-alarm rate under arbitrary drift. We compute the price of that guarantee, stating every law with its exact domain of validity. First, the guarantee is an accounting identity, insensitive to what the monitor is meant to detect. Two evidence identities make the cost exact for the online quantile tracker: a persistent step of height $\delta$ yields excess alarm mass within one alarm of $\delta/\eta$, and exactly $\delta/\eta$ pathwise when $\delta$ is a lattice multiple of the gain $\eta$; a ramp of slope $c$ yields a stationary excess rate of exactly $c/\eta$, independent of accumulated size, up to a boundary $c=\eta(1-\alpha)$ coinciding with the alarm-rate cap. Second, the certificate's own fluctuation obeys an exact law: the windowed alarm rate has standard deviation of order $1/L$, not the binomial $1/\sqrt{L}$, since the windowed mass telescopes to a difference of a tight internal state; the closed-form constant is validated with no fitted parameter. Detectors calibrated on the binomial scale are miscalibrated by $\sqrt{\eta\varphi(q_0)L}$, and correct calibration turns detection windows from quadratic to linear in the inverse fault speed. Third, any monitor required to tolerate a drift class $\mathcal{D}$ is blind, at any horizon and for any rule, to every fault in $\mathcal{D}-\mathcal{D}$; the proof is a deliberately elementary two-point argument and the contribution is the object it identifies: for speed-bounded classes the blind set is exactly the doubled-speed class, and the tracker absorbs a speed class fixed by its own gain, so that under a certification regime declaring absorbed drift normal, the monitor manufactures $\mathcal{D}$. An exact Gaussian projection bound, sharper than Pinsker and never vacuous, quantifies power outside it.

We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming–Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.

Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of $\mathbb R^d$ but require solving a transport problem, while kernel discrepancies are cheap to evaluate yet depend delicately on their test class. We study the Toscani–Fourier family $\mathrm T_{s,p}$, the weighted $L^p$ norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on $\mathbb R^d$. For $1\le p<\infty$ we show that $d/p<s<1+d/p$ is exactly the window in which $\mathrm T_{s,p}$ is finite on $\mathcal P_p(\mathbb R^d)$, both endpoints already failing for a pair of Dirac measures, and we establish the metric, embedding, and compactness structure of the resulting space, which we prove to be complete. Duality identifies $\mathrm T_{s,p}$ as an integral probability metric over a homogeneous Fourier–Lebesgue ball, with an explicit extremizer when $1<p<\infty$. We prove the global bound $\mathrm T_{s,p}\lesssim W_p^{\,s-d/p}$, whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with $W_p$ on bounded-support and uniform-tail classes, together with explicit reverse moduli on bounded-support classes that improve the imported energy-kernel exponent at $p=2$. There, $\mathrm T_{s,2}$ is a constant multiple of the classical energy distance, which yields an exact finite-sample identity for the mean of the empirical discrepancy; the numerical experiments are otherwise diagnostic.

For a finite family of real matrices, we bound the Gaussian width of the union of its image ellipsoids by its maximal Hilbert Schmidt norm, its maximal fixed input Gaussian width, and the square root of the product of its operator radius and decoupled Gaussian chaos supremum.

We give a short operator-theoretic proof of the sharp Hoeffding inequality for additive functionals of a Markov chain on a general state space with an $L^2(\pi)$ spectral gap, recovering the bound of Fan et al. (2021). We further show that the same argument yields the Hoeffding inequality of Neeman et al. (2024) for Markov-dependent random matrices. The proof rests on two key observations. First, the transition operator factors as $P=Q_\lambda^{1/2} K Q_\lambda^{1/2}$ through the León-Perron operator $Q_\lambda$, with $\|K\|=1$. This bounds the moment generating function by a product of one-step operator norms. Second, each one-step operator is a rank-one perturbation of a multiplication operator, so bounding its norm reduces to verifying a scalar resolvent condition. Scalar convexity and the classical Hoeffding lemma then give the desired estimate. This bypasses the asymptotic cumulant generating function, essential-spectrum analysis, and extremal two-state comparison used in existing proofs. In the matrix setting, the multi-matrix Golden-Thompson inequality first reduces the moment generating function to an operator problem on a lifted Hilbert space. The same two steps then apply: the projection onto constants becomes finite-rank, and a Schur-complement argument yields the same scalar estimate. The matrix Hoeffding inequality of Neeman et al. (2024) follows. For complex Hermitian summands, working directly on the complex Hilbert space improves the bound by a factor of $2$.

A right sub-gamma bound is described by a quadratic proxy $v$ and a pole $c$, and the pair is not unique: enlarging either preserves it. Fixing $v$ at the variance $V$ removes the ambiguity at quadratic order but is a convention, not a consequence. For centered infinitely divisible laws with finite nonzero variance we determine the entire boundary of the feasible set, the map $v\mapsto c_*(v)$. A Beta$(1,2)$ multiplier applied to the normalised Kolmogorov canonical measure turns feasibility into a one-dimensional comparison and yields an exact variational formula: the frontier is convex, nonincreasing, and its feasible set is convex. The abscissa of convergence of the moment generating function imposes a $v$-insensitive floor on $c_*$, so a law whose variance-exact pole sits on it has a flat frontier, whereas the third-cumulant obstruction exists only at $v=V$. Whenever that pole lies strictly above both floors the frontier drops strictly as soon as $v>V$; if the control is also purely local with $\kappa_3^2/(9V^2)>\kappa_4/(12V)$, the drop has a square-root profile, and in the remaining boundary case a cube-root profile, with derivative $-\infty$ at $V$ either way. We then characterise when the optimised frontier reproduces the exact Chernoff deviation: equality holds at a level exactly when some pair on the frontier is tangent to the cumulant generating function at a Legendre maximiser for that level. At the levels tabulated in a worked example $v=V$ costs four to thirty percent; for a centered exponential a gap persists at every level, reaching thirteen percent.

We introduce Neural $S$-Transform Reconstruction (NeSTR) for Gaussian stochastic partial differential equations (SPDEs). The learned object is not the random solution itself, but its $S$-Transform on a finite dimensional subspace of the white noise test space. For Wick type SPDEs, NeSTR yields a deterministic parametric PDE for the restricted $S$-Transform, and the stochastic solution is recovered from the Taylor coefficients of this transform at the origin. Thus the neural approximation is tied directly to the Wiener chaos expansion of the solution. The main novelty of NeSTR is an inverse $S$-Transform reconstruction principle combined with finite mode learning: we prove consistency of the recovered chaos coefficients, separate noise mode, chaos truncation, deterministic solver, and neural approximation errors, and give a deterministic learning formulation for the restricted transform. Numerical tests on an additive stochastic heat equation and a multiplicative Wick heat equation show that polynomial spectral representations in the transform variables provide stable access to higher order chaos coefficients. The resulting reconstruction separates the equation dependent deterministic coefficient fields from a reusable Gaussian–Hermite basis. Consequently, NeSTR learns an analytic generating function whose derivatives recover the Wiener chaos coefficients, rather than learning stochastic trajectories directly.

We derive representation-free densities of ratios of the positive stable random variable and obtain their Laplace/Stieltjes transforms. We state representation-free densities of occupation times (Lamperti, 1958) and Bessel process excursions (Bertoin et al., 2006) and their transforms. We also obtain simple representations of these and other densities using a known infinite series representation of the stable density.

We show that a statistical model in the sense of McCullagh, in the form given by Brøns, is a natural transformation between two functors from the category of designs to the Kleisli category Stoch of the Giry monad, provided that its components are measurable in the parameter. The condition is empty for finite models. A design-indexed quantity is a family of morphisms of Stoch defined on the parameter objects, called meaningful if it is natural. We prove that Tjur's criterion, imposed on parameter functions indexed by finite samples with multiplicities, forces the indexing by the support and then coincides with naturality over the insertions. For finite designs we show that a quantity can be corrected to a natural one within a given class of corrections if and only if a class vanishes in the first cohomology group of a Baues-Wirsching complex relative to that class, while its image in the absolute group is always zero. In the one-way layout, marginal dispersion is not meaningful, and within-group dispersion is the unique correction that leaves the merged design unchanged. A prior is a family of states on the parameter objects, called coherent over a class of design morphisms if it is natural over that class. We show that coherence at a merge confines the prior to the image of the corresponding parameter map, that coherence over the insertions is Kolmogorov consistency, and that coherence over the injections adds the exchangeability assumed by the categorical de Finetti theorem. In the finite one-way scheme, the coherent priors form polytopes of known dimension. The analogue of Jeffreys' general rule is not coherent, while the analogue for location-scale families is. Finally, we show that ridge regression is the Bayesian inversion of the Gaussian linear model with respect to a Gaussian prior, which is coherent over the insertions and never over the injections.

We give a candidate proof of Conjecture 5 in the third arXiv version of Latala and Strzelecka Operator lp lq norms of Gaussian matrices.

In this paper, we establish the existence and uniqueness of solutions for mean-field stochastic differential equations (MF-SDEs in short) and backward stochastic differential equations (MF-BSDE) driven by anomalous sub-diffusions $\{B_{L_t}; t\geq 0\}$ with random coefficients, respectively. Here $B$ is a Brownian motion on ${\mathbb R}^d$ and $L$ is the inverse of a subordinator $S$ with drift $\kappa >0$ that is independent of $B$. We further study the stochastic maximum principles (SMPs) for control problems of the stochastic systems modelled by the MF-SDEs using a convex variational method. A linear quadratic control example is given in the last section of this paper, for which both the SMP and the sufficient SMP established in this paper are utilized to show explicitly that it admits a unique stochastic optimal control.

In this paper we extend the results from Yang and Li(2017), in a non-renewal driven bivariate risk model. Concretely, we are interested in the asymptotic behavior of the joint tail of the discounted aggregate claims over finite and infinite time horizon in a bivariate risk model with two arbitrarily dependent counting processes. We additionally suppose that the sequences of claim-sizes of the two lines of business are independent, but each of them contains weakly dependent terms. In our main results, on finite and on infinite time horizon, we assume two (different) general conditions for the counting processes, that are satisfied by a wide spectrum of processes beyond the renewal ones. In the case of finite horizon we suppose that the claim distributions from the two lines, belong to the subexponential distribution class, while in the case of infinite horizon we restrict ourselves to the consistently varying and positively decreasing distribution class. More explicit asymptotic expressions are derived in the case we are restricted on the regularly varying class for the claim distributions. We note that our results indicate the presence of multivariate non-linear single big jump principle for the discounted aggregate claims.

Let \[ A_\omega=-\partial_x^2+\rho\dot B_x(\omega), \qquad \rho\ne0, \] be the Dirichlet Schrödinger operator on $(0,\pi)$ with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here $\dot B_x$ denotes the distributional derivative of Brownian motion with respect to the spatial variable $x$. For $\kappa\ne0$, we consider the cubic nonlinear Schrödinger equation \[ i u_t=A_\omega u+\kappa |u|^2u. \] We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set \(J\) of cardinality \(b\), we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant \(b\)-tori. The family is parametrized by a Cantor subset of \([\nu,2\nu]^b\) whose relative measure tends to one as \(\nu\to0\).

We prove Villani's conjecture on entropy production for Kac's walk with uniform collision angles. For every $N\geq2$, with total collision rate $N$, the optimal entropy production constant is $2/(N-1)$. We show that the known lower bound is sharp by constructing two families of smooth strictly positive probability densities, invariant under coordinate permutations and sign changes. The first consists of normalized products of inverse powers; the second is obtained by conditioning products of Gaussian mixtures on the energy sphere. With $N$ fixed, we compute the leading terms of entropy and entropy production. Successive parameter limits yield two independent proofs of sharpness.

We study the nonlinear Hamilton–Jacobi system arising as the formal semi-classical limit of the chordal Belavin–Polyakov–Zamolodchikov equations with a common eigenvalue parameter $\lambda$. We determine the number of global real-valued solutions modulo additive constants. When $\lambda=0$, the number of solutions can be derived using the connection between the solutions and rational functions, due to Eremenko. We focus on the case when $\lambda\neq 0$ in this article. For $\lambda>0$, a Schwarzian correspondence relates the algebraic equations for the gradients to quasi-exponentials with prescribed critical points. Enumeration and transversality results of Mukhin, Tarasov, and Varchenko yield smooth real branches, and a Poisson bracket identity establishes their exactness. An energy bound rules out global real-valued solutions for $\lambda<0$.

Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at $N$ equidistant times through a superposition of $m$ independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every $N\ge2$, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér–Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.

Consider the Monge-Kantorovich optimal transport problem between two Euclidean densities $\mu$ and $\nu$ and quadratic cost. The $\varepsilon$-Schrödinger bridge is the solution to the entropic regularized problem with regularization parameter $\varepsilon$. By taking the conditional expectation of the second coordinate given the first under this coupling, we obtain the $\varepsilon$-entropic Brenier map. As $\varepsilon$ goes down to zero, it is known that the entropic Brenier map converges to the quadratic cost optimal transport map between the two measures, i.e., the Brenier map. We show that, under some smoothness and log-concavity constraints on the marginals, the difference between the entropic Brenier map and the Brenier map is equal to $\varepsilon$ times one-half of the score function of the first marginal plus an error that is $o(\varepsilon)$ in $\mathbf{L}^2(\mu)$. This expansion holds irrespective of the second marginal $\nu$. The proof relies on an approximation of the Schrödinger bridge by a noisy version of the McCann interpolation. Under additional assumptions we show a second approximation to the Schrödinger bridge via so-called Mirror Langevin diffusions, which are Langevin diffusions on the Hessian manifold generated by the Brenier map. These two approximations, that appear quite different at first glance, are nonetheless shown to be closely connected. Our proofs utilize a random surface and a novel stochastic operation called the tangent Markov projection.

This paper is concerned with the Fleming-Viot particle system introduced by Burdzy, Hołyst and March (2000). In this model, $N$ Brownian particles evolve independently in a bounded domain $D$ until one of the particles reaches the boundary. Then, the particle which has hit the boundary instantaneously jumps to the location of one of the other particles, chosen uniformly at random. Burdzy, Hołyst and March showed that when $N$ tends to infinity, the empirical measure of the system converges to a solution to the heat equation on $D$ with Dirichlet boundary conditions, renormalized to have total mass $1$. Our main result is a sharp, uniform-in-time, quantitative version of this result. We employ the method of weak propagation of chaos, which necessitates a careful study of the (backward) Kolmogorov equations associated to the $N$-particle systems, and the infinite-dimensional transport equation whose characteristics are given by renormalized solutions of the Dirichlet heat equation. The proofs are entirely analytical, and most of the technical effort is devoted to building barrier functions which are used to control the singular behavior of the system when most of the particles approach the boundary.

In this paper, we consider diffusion processes on a graph. On the $i$-th edge, each process is governed by an operator of the form $D_{v_i}D^+_{u_i}$, and the behavior at the vertices is determined by the gluing conditions. The paper characterizes the weak convergence of a family of such processes to a limiting process in terms of the behavior of the scale and speed functions $u_i^n$ and $v_i^n$ and the parameters of the gluing conditions. This may be viewed as a generalization of Freidlin and Wentzell's result on the weak convergence of a family of processes on a line.

We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to $\infty$ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a $3/2$-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with $n$ edges, directed distances are typically of order $n^{1/3}$. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the $\sqrt 2$-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for $\gamma=\sqrt 2$. These results give the scaling dimension for a hypothetical directed version of the $\sqrt 2$-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.

We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,\rho)$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_\beta(x)=d_M(x,x_0)^{-\beta}$, with suitable $x_0$ satisfying $\rho(B_r(x_0))\sim C_d(x_0)r^d$, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed $\alpha$-stable laws for $\alpha=d/\beta\in(0,2)$ and, at the boundary value $\alpha=2$, a Gaussian limit with the non-standard normalisation $\sqrt{n\log n}$. We also establish a conditional central limit theorem for $L^2$ observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets.

We study variation functions of some lacunary series along the sequence of $b$-adic partitions, where $b>1$ is an integer. By such a lacunary series, we mean that in the definition of the well-known Weierstrass function, the Lipschitz continuous cosine and sine functions are replaced by a continuous and piecewise continuously differentiable periodic function whose Fourier series has zero $(jb)^{\mathrm{th}}$-Fourier coefficients for all non-zero integers $j$. In the proofs, we use a probabilistic approach applying a martingale central limit theorem. Our result extends some recent results of Han, Schied and Zhang (2021), and can be applied when one chooses the periodic function in question, for example, as the Takagi function, a Weierstrass function or a certain lacunary series itself.

We study signal prediction and smoothing in hidden Markov models whose latent signal is a multi-type Wright-Fisher diffusion and whose discrete-time observations are categorical. We consider a repeated-sampling regime with finitely many inspection times and large within-time samples. Under parent-independent mutation, the Bayesian predictive and smoothing laws admit exact representations as mixtures of Dirichlet distributions. We show that the one-step predictive and local smoothing laws converge in total variation to the corresponding Wright-Fisher transition and bridge laws determined by the neighboring population frequencies, thereby justifying plug-in approximations based on empirical proportions. For prediction, we derive explicit finite-sample bounds for this approximation and for bounded predictive functionals such as diversity, together with a finite-sample concentration bound for the marginal smoother. The joint posterior at the inspection times concentrates at the target frequencies and satisfies a Bernstein-von Mises limit with distinct interior and boundary regimes, with Gaussian and Gamma limits on different scales. The results allow arbitrary dependence across inspection times. A key technical ingredient is a fixed-interval tail bound for the coalescent-based dual death process.

Choose uniformly among the combinatorial types of convex three-dimensional polyhedra with a fixed admissible number $e$ of edges, and let $V_e$ be the number of vertices. This resolves a fixed-edge limit-distribution question posed by Rüdinger: if $\beta_e=V_e/(e+2)$, then $\sqrt e(\beta_e-1/2)\Rightarrow N(0,1/32)$, equivalently $\operatorname{Var}(V_e)\sim e/32$. Using the classical rooted enumeration and asymmetry results of Bender and Wormald, we derive a relative lattice local limit theorem on every $o(e^{3/4})$ window, a quartic correction from the rate function on every $o(e^{5/6})$ window, precise moderate-tail constants, a quadratic moderate-deviation principle for every $a_e\to\infty$ with $a_e=o(\sqrt e)$, and a full speed-$e$ large-deviation principle with an explicit good rate function. All fixed standardised moments converge. When $e=3m$, the two extremal vertex counts have equal probability asymptotic to $\frac{6561}{32\sqrt2}(4/27)^m$. Rooted and unrooted fixed-edge laws are also uniformly exponentially close, with relative discrepancy $O(\rho^e)$.

We derive explicit lower bounds for relative Fisher information by combining a variational principle with suitably orthogonalized Hermite-polynomial test functions. The resulting cumulant bounds are asymptotically sharp and yield lower bounds for Gaussian entropy deficits. We also establish quantitative versions of sharp min-entropy power inequalities in all dimensions. En route, we develop a stability result for Brzezinski's sharp bound for block sections of products of Euclidean balls, which may be of independent interest.