Papers

Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] This establishes the exponential scale lower bound suggested by Tao and Vu. The proof bounds the cumulative logarithmic loss of the sum of squared permanents of minors under row exposure.

We study high-dimensional random-matrix limits of Gaussian Kullback–Leibler unbalanced optimal transport (KL-UOT). Under equal marginal penalties, the covariance action admits an exact log-determinant representation in terms of a nonlinear ridge product, together with a positive-semidefinite extension that remains finite at arbitrary aspect ratios. For independent real Wishart samples, strong asymptotic freeness gives the limiting free multiplicative convolution and almost-sure Hausdorff convergence of the ridge-product spectrum; independent Haar orientations yield the corresponding first-order limit for deformed populations. In the symmetric nonsingular identity-Wishart model, we derive an explicit $\eta$-transform and a low-degree algebraic equation that select the physical branch and determine the support interval, square-root edges, and extreme-eigenvalue limits. We further obtain all-aspect one-sample Marchenko–Pastur limits under finite fourth moments, real-Gaussian Bai–Silverstein fluctuations for $c<1$, and a joint random-matrix/penalty limit showing that sample-covariance noise produces the critical scale $\tau_p\asymp p$.

We prove bounds of order $n^{n/2}e^{O(n)}$ for the expected number of facets of high-dimensional random polytopes. First, let $\mu$ be a non-degenerate compactly supported even probability measure on $\R$ satisfying $\mu([x^\ast-s,x^\ast])\asymp s^\kappa$ near its right endpoint $x^\ast$. For every sufficiently small fixed $\alpha>0$, the convex hull of $N=\lfloor e^{\alpha n}\rfloor$ independent points with law $\mu^{\otimes n}$ has at least $n^{n/2}e^{-C_{\mu,\alpha}n}$ expected facets; this includes all symmetric finite-alphabet distributions. For every full-dimensional log-concave probability measure on $\R^n$, we prove that there exist $T\in[n,2n]$ and $N=\lceil e^Tn^{3/2}\rceil$ for which \[ n^{n/2}e^{-Cn} \leq \mathbb E f_{n-1}(P_N) \leq n^{n/2}e^{Cn}. \] Thus the scale $n^{n/2}$, up to exponential factors, is universal for log-concave measures in this high-dimensional exponential regime. Finally, we construct a symmetric isotropic full-support non-log-concave counterexample with only $(1+o(1))2^n$ expected facets.

We consider fractional Brownian motion with D-dimensional time in the domain G(T)=(-T<ti<T,tj>0 if j<d+1). Let P(T) be the probability that the process will not exceed a fixed level in G(T). We show that for large T, logP(T)=(D-dH)logT(1+o(1)) where H is the Hurst parameter of the process

Let $(M,g)$ be a smooth, closed, connected $d$-dimensional Riemannian manifold, and let $X_1,\ldots,X_n$ be i.i.d.\ with common law $f\,d\mathrm{vol}_g$, where $f\in C^4(M)$ is strictly positive. We derive a uniform intrinsic second-order expansion for symmetric three-vertex edge-indicator statistics supported on connected configurations, including the induced-path and triangle kernels. The second-order term separates density variation, normal-coordinate Jacobians, and the curvature-induced motion of the internal-chord boundary. Within this three-vertex connected symmetric class, a universal boundary–moment identity reduces the kernel-dependent contribution to a common intrinsic functional involving $\int_M f\|\grad f\|_g^2\,d\mathrm{vol}_g$ and $\int_M f^3\operatorname{Scal}_g\,d\mathrm{vol}_g$. For the normalized path–triangle contrast, the Euclidean leading term cancels. We construct a consistent estimator of this intrinsic functional and, in a denser bandwidth regime, we prove an exact-expectation-centered root-\(n\) central limit theorem via the first Hoeffding projection. For uniform sampling on a closed surface, the estimator consistently recovers the Euler characteristic. We also study the threshold radius at which the maximum degree of a binomial random geometric graph first reaches two. Using the active-triple intensity expansion and a dependency-graph Poisson approximation, we obtain the order-$n^{-3/d}$ correction to the log-survival law for $d>6$.

We study the Kac killing penalization for a one-dimensional recurrent Lévy process. We prove that the limit process possesses two equivalent characterizations by a Fredholm equation and by a Sturm–Liouville equation.

The permanent of an $n\times n$ $0/1$ matrix $A$ equals the number of perfect matchings in the bipartite graph with edges defined by $A$. Jerrum, Sinclair, and Vigoda (2004) presented an FPRAS for approximating the permanent of any nonnegative matrix using a novel simulated-annealing algorithm. The running time was improved by Bezáková, Štefankovič, Vazirani, and Vigoda (2008) to $O(n^7\log^4 n)$ for $0/1$ matrices, for any fixed approximation and success parameters. We present the first asymptotic improvement over this running time bound, obtaining an $O(n^6\log^5 n)$-time algorithm. As in the previous works, our algorithm extends to arbitrary nonnegative matrices. The analysis of Bezáková et al. yields an $O(n^4)$ relaxation time bound for the JSV Markov chain on perfect and near-perfect matchings with ideal hole weights, under which each hole pattern (the unmatched vertices, if any) is equally likely in the stationary distribution. We introduce a restricted Poincaré inequality for the partition into hole patterns and prove an $O(n^3)$ bound on the corresponding restricted relaxation time. Our proof uses a coupled multicommodity flow argument inspired by a recent transport-flow argument of Chen et al.~(2025) for the Jerrum-Sinclair chain on all matchings.

Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subordinator; its $L^2$-generator is $-(-\Delta)^{\alpha/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/\alpha})}=\infty \] implies that \(X^{(\alpha)}\) is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-\alpha}\,d\mu\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.

We study the entropy of random graphs generated by piecewise Hölder continuous graphons. We first present a result on the rate of convergence of the normalized entropy as the size of the graph grows. The core ideas of the proof are described, with the detailed proof provided in the appendix. From this result, we then derive quantitative bounds on the entropy for the stochastic block model and random geometric graph model. These bounds provide explicit formulae rather than asymptotic statements which have been found previously.

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

We prove a periodic long-time Boltzmann–Grad limit for hard spheres in every fixed spatial dimension $d\ge4$. Under the assumptions of the main theorem, the rescaled $s$-particle correlations converge in $L^1$ to the corresponding tensor-product Boltzmann profile at rate $\varepsilon^{1/(400d)}$, uniformly for $1\le s\le|\log\varepsilon|$ and $0\le t\le t_{\rm fin}$. In particular, when the activity and weighted solution bounds are fixed, $t_{\rm fin}=O(\log|\log\varepsilon|)$. The componentwise long-bond estimate used in dimensions two and three is insufficient in higher dimension. We replace it by a joint estimate for two connected time sublayers, selecting the first two lower collisions as landing roots. Disjoint landing edges yield a direct tangential frame. When the edges overlap, the newly appearing particle line contains a collision-free chord whose radial relative speed produces a rank-$(d-1)$ positive Schur factor. A positive Jacobi-network argument prevents focusing, while bounded-degree finite-fibre coarea globalizes the estimate for arbitrary nonnegative joint kernels. A first-failure construction repairs periodic double overlaps, and the resulting packet bound closes the exceptional top-layer contribution. A paid epoch restart for sealed complete joint kernels, using the fixed-word multi-landing operator for every fixed integer $k\ge1$, yields the stated time range. Within the regular connected full two-sublayer packet class, every packet with at least $2k-1$ physical lines admits a canonical $k$-birth flag and the operator factor $\varepsilon^{(d-1)(k-1)}\varepsilon_*^{-k(d-1)}$; $k=2$ is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.

In this paper, motivated by discrete-time McKean-Vlasov optimal control problems with law-delay, a class of forward-backward stochastic difference equations is investigated. The main difficulties come from the absence of Ito's formula in the discrete-time setting, as well as the fully coupled structure in which the coefficients depend on both law-delayed and law-anticipated terms. Under suitable monotonicity conditions, the existence and uniqueness of solutions are established. Furthermore, a new monotonicity condition is introduced to address the associated optimal control problem. As an application, the unique optimal control for linear quadratic systems with law-delay is derived.

Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+\lambda/n$ in dimensions $n$ and $n-k$. Uniformly over deletion ranks retaining a fixed positive fraction of observations, the first nonzero difference converges to an explicit phase surface $H_\delta(\lambda)$; its zero curve unifies fixed, sublinear and fixed-fraction deletions, with tangent crossing $12/35$. Through the Khintchine scale-mixture representation, this comparison yields a single compactly supported $C^\infty$ symmetric unimodal parent, independent of $n$, whose Student tail exceeds the equal-scale uniform tail for all sufficiently large $n$ along nominal levels tending to $2\{1-\Phi(\sqrt3)\}$ from below. In contrast, a quantile-ratio order shows that the uniform parent maximizes every even moment and every convergent even power series with nonnegative coefficients. We also derive the fixed-confidence dimension expansion and an exact reversal at $n=6$.

We consider a non-interacting Bose gas governed by a general one-particle Hamiltonian in the canonical ensemble. Our main results are large-deviation estimates for the number of particles in the Bose–Einstein condensate. We show that the left- and right-tail probabilities decay at different exponential rates as the number of particles tends to infinity.

We show a birthday paradox for random non-backtracking walk on regular graphs of degree at least $3$: such a walk of length $k$ has high probability of self-intersecting when $k$ is significantly greater than $\sqrt n$, where $n$ is the number of vertices of the graph. This resolves a conjecture of Noga Alon and Yuval Peres for the fixed degree case.

We prove cutoff with an $O(1)$ window for the continuous-time heat-bath Glauber dynamics of the ferromagnetic $q$-state Potts model on the discrete torus $\Lambda_n=(\mathbb Z/n\mathbb Z)^d$ at sufficiently high temperature. For every fixed $d\ge2$ and $q\ge3$, there exists $\beta_0=\beta_0(d,q)>0$ such that, for $0<\beta<\beta_0$, the Glauber dynamics of the Potts model on $\Lambda_n$ exhibits cutoff with optimal $O(1)$ window around \[ t_\star=t_\star^{(n)}:=\frac{1}{2\mathfrak{r}}\log |\Lambda_n|, \] where $\mathfrak{r}\in(0,1)$ is the exponential decay rate of the one-site magnetization. In particular, this determines the mixing time up to an additive $O(1)$. It is characterized by the point at which the macroscopic color-density bias from the monochromatic initial condition enters the scale of equilibrium fluctuations. Moreover, our proof shows that the monochromatic initial condition uniquely maximizes the color bias. This is the first implementation of information percolation to prove cutoff for a non-monotone spin system. In contrast with the Ising model, a direct implementation of information percolation does not yield matching upper and lower bounds for the Potts dynamics when $q\ge3$. We overcome this by developing an information-percolation framework for signed influences and combining it with Fourier bounds on signed convolution powers and geometric control of history diagrams.

For matrices drawn from the standard Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE), we prove that the normalized zero counting measure of the permanental characteristic polynomial $Per(zI_N-H_N)$ converges almost surely to the standard Wigner semicircle law on $[-2,2]$, rotated by $\pi/2$ onto the imaginary axis. This proves the conjecture proposed by Fyodorov in \cite{Fyodorov2006}.

Since Hastings' proof of superadditivity of classical communication over quantum channels, considerable effort has been devoted to finding a structural explanation of this phenomenon that was originally established by concentration of measure for Haar random unitaries. The main observation of this work is that Haar randomness can be replaced by random permutations without changing the limiting geometry responsible for nonadditivity. This replacement turns a continuous problem over unitary matrices into a discrete combinatorial problem over zero–one permutation matrices, and thereby opens a path toward derandomization. The theorem of Bordenave and Collins shows that random permutations have the required limiting behavior and the algorithm of O'Donnell and Wu then provides a deterministic asymptotic construction, running in polynomial time in the size when the channel parameters and accuracy are fixed. Thus the random construction can be derandomized in an asymptotic algorithmic sense. Finally, a quantitative random permutation estimate by Chen, Garza-Vargas, Tropp and van Handel gives a fully numerical estimate: there exists a tuple of 57,836,025 permutations acting on a set of size \[ N \le 5.422\times 10^{116216}\] such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.

We study a class of discrete-time interacting particle systems arising from mean-field coupled maps. We establish a Dobrushin-type estimate, a relative entropy estimate and Central Limit Theorem (CLT) type results for this class of systems. First we control the growth of the 1-Wasserstein distance between the empirical measures and the limiting distributions. Then we use the relative entropy method to show the propagation of chaos. Finally we consider the asymptotic behavior of the fluctuations for the empirical measures and prove that the sequence of fluctuation processes converges in distribution to some Gaussian process, where we establish both qualitative and quantitative results.

We propose and analyze a first-order numerical scheme for a class of scalar Itô stochastic differential equations with almost surely positive solutions. We construct a new explicit numerical scheme, such that for any choice of the time-step size, the numerical solution is guaranteed to remain almost surely positive. The main result of this article is the first-order strong convergence of the proposed positivity preserving scheme, which is illustrated with numerical experiments and proved rigorously. It is also shown how to adapt the scheme and the results to Stratonovich stochastic differential equations with positive solutions.

We construct and analyze numerical schemes for systems of stochastic differential equations, which preserve almost surely a given hypercube of arbitrary dimension. We propose a new general class of explicit schemes, such that for any choice of the time-step size the numerical solution takes values in the hypercube. We prove strong and weak convergence results for this general class of domain preserving numerical schemes, with strong order $1/2$ and weak order $1$ in general. We also construct a variant of the scheme which achieves strong order $1$ when the stochastic differential equation is driven by a one-dimensional Brownian motion. The convergence results are illustrated with numerical experiments.

It is proved that, for every $\delta \in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+\delta)\Delta$ and the underlying graph has girth at least $5$ and maximum degree $\Delta = \Omega_\delta(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+\delta)(1-\beta)\Delta$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

We study the fine geometry of collision measures associated with independent Markov processes on a common metric measure space. These measures are natural space-time random measures encoding collision times and collision points. Under small-scale Ahlfors regularity and standard short-time heat-kernel estimates, we determine the local dimensions of their temporal and spatial marginals, the Hausdorff dimensions of their supports and of the collision sets themselves, and logarithmic limsup laws for their local masses at typical collisions and at exceptional thick collisions. Through this analysis, we identify exact Hausdorff measure functions for the collision-time sets and, in a natural regime, for the collision-point sets. Moreover, we prove that the corresponding Hausdorff measures are comparable, uniformly over all Borel subsets, to the temporal and spatial marginals of the collision measures. This provides a broad solution to the Hausdorff-measure part of an open question posed by Xiao more than two decades ago. Our framework covers symmetric stable processes, canonical diffusions on affine nested fractals such as the Sierpiński gasket, and stable-like jump processes on $d$-sets.

We study the Sherrington-Kirkpatrick model with an additional Curie-Weiss ferromagnetic interaction (SKFI), whose phase diagram in the plane of the disorder strength $\beta$ and the ferromagnetic coupling $\gamma$ consists of a paramagnetic, a ferromagnetic, and a spin glass region. Our main result is a central limit theorem for the free energy in the ferromagnetic regime: at scale $N^{-1/2}$, the fluctuations coincide with those of the Sherrington-Kirkpatrick model in an effective external field determined by the limiting magnetization. We prove this for general mixed even $p$-spin interactions, using the fluctuation theory of Chen, Dey, and Panchenko. For the pure 2-spin model we complete the picture of the phase diagram: we give a new proof of the order-$N^{-1}$ Gaussian fluctuations in the paramagnetic regime, obtained earlier by Banerjee, valid up to the critical window; we determine the limiting distribution on the critical line $\gamma = 1 + \theta N^{-1/2}$ separating the paramagnetic and ferromagnetic regimes; and we show that in the spin glass regime the fluctuations coincide with those of the zero-field SK model under a widely believed variance-divergence assumption. Our results form the Ising analogue of the fluctuation results of Baik and Lee for the spherical SKFI model.

Exponential random graph models (ERGMs) describe dependence among network ties, but inference becomes difficult when the likelihood is intractable and candidate network statistics are strongly correlated. We introduce BERGM Elastic Net, an adaptive empirical-Bayes approach that combines lasso shrinkage with ridge stabilization in a Bayesian ERGM. A latent-variable formulation supports approximate exchange sampling, while empirical-Bayes updates adapt the amount of regularization to the observed network. We connect the proposed prior to elastic-net penalized likelihood and clarify the interpretation of thresholded reporting and coefficient grouping. The method is developed for over-specified network models containing many related structural and covariate effects.

The OBABO and BAOAB schemes and the other standard Strang splittings of kinetic (underdamped) Langevin dynamics are widely used Markov chain Monte Carlo algorithms. Under a suitable friction scaling, the underlying diffusion relaxes on a ballistic time scale, suggesting that these discretizations, suitably tuned, sample targets with condition number $\kappa$ in $O(\sqrt{\kappa})$ iterations. We prove that no fixed choice of step size and friction, based only on the curvature bounds and the dimension, achieves this acceleration: total variation mixing time lower bounds for OBABO show that ballistic cold-start mixing fails uniformly over the smooth strongly convex class, and the lower bounds extend, with the same orders, to BAOAB and the other four Strang splittings. The proof transfers non-acceleration from optimization to sampling. Eliminating velocity gives an exact noisy heavy-ball recursion, and by the non-acceleration theorem of Goujaud, Taylor and Dieuleveut, for every tuning either some Gaussian target has a mode with relaxation time at least of order $\kappa$, or an attracting cycle exists on a smooth potential; dilating such a potential as $U_R(x)=R^2U(x/R)$ preserves its curvature bounds and produces metastability for a number of steps exponential in $R^2$, from an initial state at Wasserstein distance $O(R)$ from equilibrium. Using contraction estimates of Leimkuhler, Paulin and Whalley and a Wasserstein-to-total-variation regularization estimate, we prove a complementary upper bound of $O(\kappa)$ steps, up to logarithmic factors, for a fixed-parameter OBABO tuning. Hence, among fixed-parameter OBABO tunings, the optimal condition-number dependence of cold-start total variation mixing over this class is linear, up to logarithmic factors. A direct Gaussian calculation also rules out fixed-parameter acceleration for the left-endpoint exponential integrator.

We develop a PDE-based methodology for pricing and hedging European contingent claims in general one-dimensional diffusion markets characterized solely by their scale function and speed measure, possibly without a classical SDE representation, and with constant interest rate. We derive a hedging equation whose solution generates a self-financing hedging strategy and provide sufficient conditions on scale, speed, and interest rate, under which this strategy achieves the minimal hedging capital, expressed through the no free lunch with vanishing risk (NFLVR) condition. We further prove necessary and sufficient conditions for NFLVR and characterize the class of equivalent local martingale measures through an auxiliary diffusion whose scale and speed characteristics are determined by those of the real-world diffusion and by the interest rate. When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital. We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features.

We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional spatial scale-free graphs whenever the standard strong-decay long-edge estimate holds. In dimension two, we combine the linear chemical-distance estimate of Lüchtrath with an area-order bound on the degree measure. Graph-distance layers in exponentially separated bands then give the required Nash-Williams cutsets for planar random geometric graphs satisfying the polynomial mixing and long-edge estimates [J. Theoret. Probab. 39 (2026), Paper No. 12]. As a concrete consequence, every connected component of the two-dimensional weight-dependent random connection model with interpolation kernel is recurrent throughout the strong-decay region $\delta>2$, $\gamma<1-\frac{1}{\delta}$, and $\alpha<1-\gamma$.

We study a linear filtering problem in which the signal process is described by a time-fractional linear stochastic differential equation driven by Brownian motion. We derive a stochastic integral equation for the conditional mean alongside a Riccati–Volterra type integral equation for the mean-square error function. As a core application, we introduce a time-fractional state-estimation framework for modelling learning trajectories in children with developmental dyscalculia.

In this note, we establish a connection between denoising diffusion models and large temperature dynamic Schrödinger bridges computed with respect to certain reference processes. In particular, we show that in the large temperature regime (i.e. $T \to +\infty$), the Schrödinger bridge behaves like a forward time diffusion process over the time interval $[0,T/2]$, and approximately like the time reversal of a diffusion process over $[T/2,T]$. This construction provides a process-level characterization of the gradient flow approximation developed by Clerc et al. (2023).