Papers

We study Open Problem 3 of Kolokoltsov's survey "Quantum filtering and propagation of chaos for open quantum systems" (2026) on $H=\mathbb{C}^d$ ($2\le d<\infty$ fixed): a dynamic central limit theorem for $N$ mean-field coupled quantum particles under continuous diffusive (homodyne) measurement. The tagged conditional state $\Gamma^{(1)}_t$ is compared with the one-particle filter $\gamma_t$ driven by its own innovation, the mean-field Hamiltonian evaluated at the deterministic Hartree-Lindblad mean $\eta_t$. The fluctuation $F^N_t=\sqrt{N}(\Gamma^{(1)}_t-\gamma_t)$ is not autonomous in the limit: it is driven through the linearised filter by its own innovation, by an idiosyncratic Gaussian martingale carried by the connected correlations with the other particles, and by the empirical field $\hat G^N_t=\sqrt{N}(\Gamma_t-\eta_t)$, which itself satisfies an autonomous linear Gaussian equation, so that $(F,\hat G)$ solves a closed system. Unconditionally, $(F^N,\hat G^N)$ is tight, every limit point solves this system driven by a Brownian motion orthogonal to a square-integrable martingale pair, and the mean covariance of the idiosyncratic martingale is identified as the limit of the rescaled pair correlations. The remaining clauses, concerning signs rather than amplitudes, reduce to a single decorrelation statement fixing the law of the driving pair. It survives in a conditional form, with the common response to the empirical field and the tagged-slot component projected out; without those projections it is false for generic data, which is what fixes them. The conditional form is proved at leading order on a shrinking horizon and from below at its exact orders on every horizon; it is needed only for the concentration of the driving covariances, the Gaussianity of the pair, and uniqueness, and what remains open is one-sided and above the leading order.

Stochastic gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of stochastic gradients by Gaussian noise, providing a powerful tool for understanding its dynamics and long-time behavior. We investigate whether an analogous approximation principle holds for optimization over probability measures, where the objective is a functional defined on the Wasserstein space P2. The nonlinear geometry and infinite-dimensional nature of P2 prevent a direct extension of the classical Euclidean theory. Using Lions differentiability, we lift the problem to a linear Hilbert space, where higher-order differential calculus becomes available. We then construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. By exploiting this moment matching through higher-order Taylor expansions, we show that the Gaussian approximation captures the SGD dynamics with second-order weak accuracy. Our result provides a rigorous foundation for replacing sample-driven randomness by analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $\lambda(s)=\log\{\Gamma(1+s)/\Gamma(1+s/2)^2\}$, let $s_\kappa$ minimize $(1+\kappa\lambda(s))/s$ on $(0,\infty)$, and put $v(\kappa)=\kappa\lambda'(s_\kappa)$ and $a_\theta=\lambda(s_\theta)/s_\theta$. Under the Ewens measure with any fixed parameter $\theta>0$ we prove $M_n=v(\theta)\log n+a_\theta(K_n-\theta\log n)+O_P(\log\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $\theta$: for every compact $[\kappa_-,\kappa_+]\subset(0,\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $\kappa_-\log n\le k\le\kappa_+\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.

We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $\lambda I$ (where $\lambda$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $\beta\leq \tfrac{1}{2}+\varepsilon$, for an absolute constant $\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.

We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size $n$, vertices are added sequentially, with the $i$th vertex starting a new branch with a prescribed probability $f(n,i)$; otherwise, it extends the current branch. We also define a continuum analogue by replacing the Poisson point process of intensity $tdt$ in the construction of the Brownian continuum random tree by one of intensity $f(t)dt$. We establish scaling limits for two families of discrete aggregation trees in the Gromov-Hausdorff-Prokhorov topology. When $f(n,i)=(i/n)^\beta$, with $\beta>0$, after rescaling the graph distance by $n^{-\beta/(\beta+1)}$, the random tree converges to the compact aggregation tree with $f(t)=t^\beta$. This recovers convergence to the Brownian continuum random tree when $\beta=1$ (Aldous, 1991), as well as scaling limits of choice spanning trees for integer $\beta$ (Archer and Shalev, 2024). We also prove convergence under rescaling to the compact aggregation tree with $f(t)=\log^\gamma(1+t)$ for every $\gamma>1$. Finally, we identify necessary conditions for compactness. Consequently, the threshold $\gamma>1$ in the logarithmic family is sharp. These results provide insight into an open problem on compactness criteria for random aggregation trees (Curien and Haas, 2014).

In this article, we prove that the solution of the skew stochastic heat equation converges to the solution of the stochastic heat equation with reflection in the limit of infinite skewness. The result holds at equilibrium. This provides an SPDE analog of the link between skew and reflected Brownian motion. One of the intermediate results is the convergence in law of a penalised Brownian bridge to a 3-Bessel bridge.

We extend reflection coupling for uniform-in-time propagation of chaos from Euclidean space to complete Riemannian manifolds. Intrinsic reflection along minimizing geodesics replaces the Euclidean difference process, and the associated endpoint-index term quantifies the competition between curvature, confinement, and interaction. Under an endpoint-index bound and a radial Laplacian bound, we establish global well-posedness, uniform moment estimates, and uniform-in-time propagation of chaos with rate $N^{-1/2}$ in a modified Wasserstein distance. We further obtain exponential contraction of the nonlinear McKean–Vlasov semigroup, uniqueness of its invariant probability measure among probability measures with finite second moment, and exponential convergence to equilibrium. Although a Ricci lower bound is an important special case, our framework also includes a curvature-spike class with $\inf_M\operatorname{Ric}=-\infty$.

Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\log\log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.

We employ a Gaussian convex inequality developed in [AC15], [Che26], [Gue22], and [FUMPC26] to establish (one sided) moment comparison inequalities for Gaussian multiplicative chaos, directed polymers in Gaussian environment and, in particular, the Critical 2d Stochastic Heat Flow (SHF), including upper bounds on negative moments, as well as similar lower bounds for $p$-th moments with $p\in (0,1)$. For the Critical 2d SHF, averaged over small balls, we also obtain the matching (up to multiplicative constants) complementary bounds. We establish the complementary bounds using a bootstrap argument inspired by [DS10] and a geometric decomposition introduced in [GT26]. Specifically, we prove that for locally averaged SHF, for every $p\in\mathbb R$, the $p$-th moment is, up to multiplicative constants, bounded in both sides by the $\frac{p(p-1)}{2}$-th power of its second moment.

We establish large deviation principles both for the CRT in the Gromov-Hausdorff topology and for the CRT weighted by its mass measure in the Gromov-Hausdorff-Prohorov topology. In the first case, the rate function of a tree $\mathcal{T}$ is half the square of the length of $\mathcal{T}$. In the second case, the rate function of a pair $(\mathcal{T},\mu)$ is half the integral with respect to the length measure of $\mathcal{T}$ of the inverse of the absolutely continuous part of $\mu$.

We study weakly interacting bosons on the three-dimensional torus at temperatures proportional to the critical temperature for Bose–Einstein condensation. We show that, as the expected particle number tends to infinity, the grand canonical Gibbs state is asymptotically described by the coherent state quantization of a random field that converges to a novel Bogoliubov random field. Moreover, the point process associated with the Gibbs state can be approximated by a Cox process. These results are based on and extend the approximation of the Gibbs state recently obtained by the first two authors and Nam. Using these approximations, we compute limiting distributions of the number of particles outside the condensate, the joint occupation statistics of finitely many momentum modes, and particle-number fluctuations in macroscopic subsets of the torus, all governed by the Bogoliubov field. We further prove convergence of the empirical measure of the associated point process to the uniform measure and convergence of the microscopic particle-number statistics to the boson point process. The latter result establishes the universality of particle-number statistics on microscopic length scales.

Let $P=(P_j)$ be a random mass partition and let $(X_j)$ be iid real marks, independent of $P$. We construct a fixed mark distribution for which the law of the real random variable $\sum_j P_j X_j$ determines the law of $P$ on the Kingman simplex. The observation map is an affine topological embedding. One construction uses an infinite convolution of stable laws with rationally independent indices; for every prescribed $q\in(0,\infty)$, a determining mark can instead be chosen symmetric, centered, compound Poisson, and in $L^q$. At any known positive time, the law of the ranked block frequencies of a $\Xi$-coalescent started from singletons$\unicode{x2014}$equivalently, the exchangeable partition probability functions of all its finite restrictions at that time$\unicode{x2014}$determines the finite collision measure $\Xi$. Composing the two results therefore identifies $\Xi$ from the law of one real marked sum. Within the $\Lambda$ subclass, an asymmetric Bernoulli coloring determines the normalized collision measure, as does any centered nonconstant mark whose absolute moment of some order $a\in(1,4)$ is finite while every higher absolute moment is infinite. We also prove that, for every $d\ge 2$, a pair of distinct laws supported on strictly decreasing $(d+1)$-tuples of positive masses with a fixed total mass remains indistinguishable for every mark supported on at most $d$ points. For every fixed number $d\ge 2$ of types, this obstruction yields distinct normalized collision measures with the same mutation-free neutral $d$-type $\Xi$-Fleming$\unicode{x2013}$Viot transition semigroup.

Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials $H_{n,m}(z)=p_n(z)+\overline{q_m(z)}$ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is $O(n)$ when $\deg p=\alpha\deg q$, where $0\leq\alpha<1$. This result extends the previous estimates when $m$ is a fixed constant or $m=n$ to more general case.

We establish Berry-Esseen bounds for the number of real roots of Gaussian Weyl polynomials $P_n$, where $n$ denotes the degree and is assumed to be sufficiently large. For each fixed $B$ above an absolute threshold, let $I_n=[-\sqrt n+B\sqrt{\log n}, \sqrt n-B\sqrt{\log n}]$. Uniformly over deterministic compact intervals $I\subseteq I_n$ whose length $\ell$ is sufficiently large and depends on $n$, the distribution of the standardized number of real roots in $I$ has Kolmogorov distance at most $C\log\ell/\sqrt\ell$ from the standard Gaussian distribution. Consequently, every such interval sequence with $\ell\to\infty$ satisfies a central limit theorem. In particular, taking $I=I_n$ gives the bound $C\log n/n^{1/4}$. The same bound holds for the distribution of the standardized number of real roots on $\mathbb R$. The key idea is to approximate the polynomial zero count by a sum of locally dependent random variables. We first couple the polynomial to a stationary Gaussian process and then truncate a moving-average representation of that process to obtain finite-range dependence. This strategy provides a route to Berry-Esseen bounds for other random polynomials with Gaussian coefficients whenever such a stationary approximation and quantitative truncation are available.

The asymptotic behavior of the coefficients of a sequence of polynomials is given under the conditions that all the zeros are real and positive and the zeros have an asymptotic zero distribution on $(0,\infty)$. We show how the limit for the coefficients is related to the Stieltjes transform of the asymptotic zero distribution and make a connection with the $S$-transform in free probability. We illustrate with some examples involving orthogonal and multiple orthogonal polynomials.

We prove that minimal-volume hyperplane sections of the regular simplex through its centroid are parallel to a facet. The proof combines variational methods with Fourier-analytic techniques and zero-diminishing arguments to show that every critical normal vector has at most three distinct non-zero coordinates. Analysis of the two- and three-value cases then yields the sharp lower bound.

We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}\beta_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\in L^\infty(\Omega;L^2(0,T;X))$, where $\beta$ is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator $A$ has sectorial angle $\pi/2$ and a bounded $H^\infty$-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an $L^2$-maximal estimate on $\mathbb R_+$ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary $C_0$-semigroups under the stronger condition $g\in L^2([0,T]; L^\infty(\Omega;X))$.

Mean field control games (MFCGs) provide a framework for studying non-cooperative games involving groups of cooperative players in the limit as both the number of players and the number of groups go to infinity.This class of games includes mean field games (MFGs) and mean field control (MFC) problems as special cases, corresponding to purely non-cooperative and purely cooperative settings, respectively. We incorporate interactions through the mean of actions, hence the terminology of extended MFCGs. We first show that the equilibrium control can be described using a coupled system of partial differential equations consisting of a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation. After introducing the general framework, we focus on a linear-quadratic (LQ) structure. We show that the equilibrium can be reduced to a system of ODEs, generalizing those obtained for MFGs and MFC problems. We then provide two numerical examples illustrating specific features of MFCGs.

In this note, we propose a general structural approach to model a default time $\tau$ as the first-passage time (FPT) of a (“firm-value”) process $S$ below a (“debt”) barrier $K$ that comply with a pre-specified survival probability curve $G(t)=\Pr(\tau>t)$. Following an idea of Mbaye and Vrins (Mathematical Finance, 2022) applied to reduced-form models, our approach consists in two steps: choose a latent FPT model driven by a barrier $\tilde{K}$ and process $\tilde{S}$, and time-change those using a deterministic clock $\Theta$ to get $K_t=\tilde{K}_{\Theta(t)}$ and $S_t:=\tilde{S}_{\Theta(t)}$, leading to the final FTP model $(K,S,\Theta)$. As the market curve $G$ and the latent model $(\tilde{K},\tilde{S})$ are assumed to be given, the calibration step simply consists in finding the clock $\Theta$ such that the distribution of the FPT of $S$ below $K$ coincides with the survival curve $G$. We show that this is achievable for a broad class of specified curves $G$ and latent FTP models. The calibration amounts to a simple inversion of a function, which is almost immediate provided that the latent model is tractable enough. In particular, we show that the AT1P model of Brigo, Morini and Tarenghi - which is able to reproduce a broad range of CDS term-structures - can be regarded as the FPT of a time-changed drifted Brownian motion to a constant barrier: $\tilde{V}_t=\mu t+W_t$ and $\tilde{K}_t=k<0$. This connection offers an elegant interpretation for the instantaneous volatility function featured in AT1P and yields an immediate calibration of the latter to perfectly match a target survival curve.

Finding large lattice subgraphs in sparse Erdős–Rényi random graphs is a classical problem at the interface of random graph theory and algorithms. General bounded-degree embedding and universality theorems give powerful results for broad graph families, but when specialized to square grids they operate at densities substantially larger than the grid-emergence scale. In this paper we exploit the specific geometry of the square grid. We introduce the Quarantined Local Search, a three-phase local algorithm that separates the construction of an initial boundary from the later corner-closure process and controls adaptive negative exposure through bounded pair-test histories. We prove that, for every fixed $\delta\in (0,1)$, there exists $C_\delta>0$ such that the algorithm embeds a $k\times k$ square grid with $k^2\le (1-\delta)n$ in $G(n,p)$ with high probability whenever $p\ge C_\delta\sqrt{\ln k/n}$. Thus, for $k^2=\Theta(n)$, a density of order $\sqrt{\log n/n}$ is sufficient, a factor of order $\sqrt{\log n}$ above the corresponding $n^{-1/2}$ emergence scale.

The paper studies finite-horizon ruin for an insurance company with several lines of business admitting heavy-tailed claims. The claim amount process of the insurance company is modeled as a multivariate increasing Lévy process possibly perturbed with fluctuations. A relevant insolvency event may involve the failure of a single line, of several lines at once, or of groups of lines linked through internal transfers, reinsurance, or a guarantee arrangement. These events have different geometries and need not occur on the same probability scale. Under asymptotic tail independence among claims from different lines, classical regular variation assigns zero limiting probability to a multi-line failure and is therefore unable to identify either the ruin probability or the corresponding solvency capital requirement. Using a multivariate Lévy process to model cumulative claims and regular variation on a nested sequence of subcones of the positive orthant to model dependence across lines, we provide the exact scaling rate of the ruin probability and the limit behavior. Additionally, explicit computations are presented for examples with independent business lines, common arrivals of independent claims, Gaussian copula dependent claims, Marshall-Olkin dependent claims, and bipartite insurance networks. Finally, we use Monte Carlo simulations to support our theoretical findings.

We establish the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in every fixed dimension $d\geq1$. For $\mu$-strongly convex, $L$-smooth potentials with unknown minimizers $x_f^\star$ in the ball $\mathbb{B}(0,\mu^{-1/2})$, and unbiased gradient oracles with variance at most $\sigma^2$, the minimax worst-case expected query complexity is $\Theta\left(\log(1+\kappa)+\frac{\sigma^2}{\mu\varepsilon}\right)$, jointly optimal for the condition number $\kappa:=L/\mu$, variance $\sigma^2\ge 0$, and TV accuracy $0<\varepsilon\leq 1/10$. In the noiseless setting $\sigma=0$, this tight complexity $\Theta\left(\log(1+\kappa)\right)$ is independent of $\varepsilon$. Moreover, a gradient-only sampler generates an exact sample with ${O}(\log(1+\kappa))$ worst-case expected queries. Without previous ball $\mathbb{B}(0,\mu^{-1/2})$, exact sampling from any initial point $x_0$ can be implemented with expected cost ${O}\left(\log(1+\kappa)+\log(1+\sqrt{\mu}\|x_0-x_f^\star\|)\right)$, without knowing the initial distance. We also show that dependence on initial distance is generally unavoidable.

Motivated by a within-host framework of immunity-modulated parasitic disease, we formulate a multitype Markovian branching process with reproductive parameters dependent on the number of individuals of a single type only. We impose a soft carrying capacity $K$ with respect to the controlling type, serving as a threshold between subcritical and supercritical dynamics. We prove that extinction occurs almost surely when the offspring batch of each non-controlling type includes an individual of the controlling type with positive probability, with the time to extinction possessing finite moments of all orders. We then construct a sequence of density-dependent processes indexed by $K$ and study scaling limits of the process using the canonical functional law of large numbers (FLLN) and central limit theorem, with a view towards characterising the emergence of metastable behaviour in the vicinity of asymptotically stable equilibria, or stable limit cycles of the FLLN.

We give a self-contained proof of Bourgain's finite $\Lambda(p)$ selection theorem with an explicit probability bound. For every $p>2$ and $N\ge2$, given $N$ pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality $\lceil N^{\frac{2}{p}}\rceil$ satisfies the $\Lambda(p)$ inequality with probability at least $1-\frac{1}{N}$, with a constant depending only on $p$. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed $A>0$, the success probability is at least $1-\frac{1}{N^A}$ with a constant depending only on $p$ and $A$, and independent of $N$. The proof uses a greedy approximation in $L^r$, with $r>2$, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.

This note presents new findings on the distribution of the volume and surface area of both the Poisson polyhedron and the Crofton polyhedron. It also considers the distribution of the area of the Poisson polygon and the Crofton polygon in the plane. This note motivates the modeling of the distributions of these convex body functionals as mixture distributions and the essential inclusion of the generalized gamma distribution therein as mixture components. Finally, model parameters are estimated and models are selected based on extensive stochastic simulations of these random polytopes. This includes that for the first time, this note gives very accurate estimates of the side number probabilities for both the Poisson polyhedron and the Crofton polyhedron, as these are used as mixing weights in the distribution models.

This article establishes sharp strong error estimates for the fully discrete approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with the linearly implicit Euler scheme in time. By mapping the diffusion operator into negative fractional spaces, our framework accommodates both trace-class and space-time white noise. For nonsmooth initial data, by decoupling the noise regularity parameter $\beta \in (0,2)$ from the initial data regularity parameter $\mu \in (0,2]$, we derive sharp regularity estimates that isolate the exact loss of initial regularity into an integrable temporal singularity. Furthermore, we establish sharp strong convergence rates of $O(h^{\beta-\varepsilon} + k^{\frac{1}{2}\min\{\beta-\varepsilon, 1\}})$ for $\varepsilon>0$ away from $t = 0$. Finally, we consider physically relevant stochastic models, such as the modified Langmuir fractional surface coverage model and the parabolic Anderson model, in our numerical experiments to confirm the theoretical convergence rates.

We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy–Littlewood maximal operator for every $1<p<\infty$, proving that it equals $p/(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly larger than the constant associated with the critical power $|x|^{-1/p}$, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.

We develop a framework for deterministic approximate counting of multi-spin systems beyond bounded-degree graphs. The algorithm recursively constructs rational polytopes containing the true marginal vectors and uses linear-fractional programming to obtain certified bounds on marginal ratios. For positive interactions on graphs of polynomial connective constant $D$, we establish strong spatial mixing and a fully polynomial-time approximation scheme (FPTAS) whenever $Dc<1$, where $c$ bounds the Birkhoff contraction coefficients of the interactions. We further extend the framework to proper colorings of sparse Erdős-Rényi random graphs using recursion on permissive blocks. For every fixed $\eta\in(0,1)$, sufficiently large fixed $d$, and fixed integer $q\ge(2+\eta)d$, we obtain an FPTAS for counting proper $q$-colorings of $G\sim\mathcal G(n,d/n)$ with high probability over $G$. This improves the leading constant $3$ in the earlier counting guarantee of Yin and Zhang (APPROX/RANDOM, 2016) to $2$, and asymptotically matches the spatial mixing regime established by Yin (ICALP, 2014).

We investigate the asymptotics of the expected infimum of log-normal Brown-Resnick stationary processes, a class of processes that arise naturally in the study of extremes of Gaussian processes and max-stable processes. Specifically, we analyse the functional $$\mathcal{G}_V(T) = \mathbb{E}\left\{\inf_{t \in [0,T]}e^{ \sqrt{2}V(t)-\sigma^2_V(t)} \right\}, \quad T>0, $$ where $V$ is a centered Gaussian process with stationary increments, continuous sample paths and variance $\sigma_V^2$, and a closely related problem of the decay rate of the persistence probability $$p_V(T,C)=\mathbb{P}\left\{\inf_{t\in [0,T]} (\sqrt{2} V(t)- \sigma^2_V(t)) > C\right\}$$ for some constant $C<0$. For both $\mathcal{G}_V(T)$ and $p_V(T,C)$ we derive exact asymptotics as $T \to \infty$ for a broad class of processes $V$, including fractional Brownian motion with Hurst parameter $H \in (1/2,1]$. For the latter, the behavior in the short-range dependence regime $H \in (0, 1/2]$ is markedly different and, in general, more delicate; we find logarithmic asymptotics for a family of processes that includes this case. Our results provide sharp bounds and comparison principles, and highlight the contrast between the behavior of infimum and supremum functionals for such processes. The discrete-time analogues and connections to Pickands constants are also discussed.

We study the overhand shuffle with a fixed cut probability $p\in(0,1)$. In the packet-reversal convention, each gap is cut independently with probability $p$, and each resulting packet is reversed in place. We prove total-variation cutoff at \[ \frac{p^2}{2(1-p)\pi^2}\,n^2\log n. \] The lower bound uses an exact finite-deck cosine eigenfunction and a uniform conditional-variance estimate. For the upper bound, a coalescing coupling reduces a transposition discrepancy to a symmetric killed two-card chain. Its principal decay rate is $(1-p)\pi^2/(p^2n^2)+o(n^{-2})$. A boundary-flux identity bounds the sum of adjacent discrepancy probabilities without a factor equal to the deck size. The remaining ingredient is a two-sided $L^1$ inequality on permutations, with constant at most $\sqrt n\exp\{O(\sqrt{\log n\log\log(n+2)})\}$. Its proof decomposes a fixed block-table fiber into assignment words and independent smaller permutations, and closes a recursion using a multivariate hypergeometric inequality and an explicit routing comparison. We also prove relative-entropy cutoff with an $O_p(n^2)$ window about the intrinsic entropy transition time. Its leading coefficient lies between the total-variation coefficient and twice that value; a sharp entropy location is not obtained. The proof states the normalization and hypotheses of its established functional-inequality inputs. An appendix proves the intermediate static bounds and explains the change from one-scale averaging to recursive fibers.