Papers

We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra–Itzykson–Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices $A,B$ with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for $(BA)^*(BA)\otimes\mathbf{1}_N$ has two limiting descriptions at speed $N^2$ for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for $(A^*A)\otimes(B^*B)$. For every fixed number of factors $k\geq3$, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number $k$ of factors, global Haar vectors at speed $N^k$ and product-state vectors at speed $N$ yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the $S$-transform.

The filtering problem asks to estimate a signal process, the initial state of which is unknown, given its partial and noisy observations. From a probabilistic perspective, the best estimate at each time is the optimal filter, which is the conditional law of the signal given all observations up to the time. The problem of asymptotic stability of the optimal filter, which asks whether the filter initialized with the incorrect initial law converges to the correctly initialized filter, is of fundamental practical and theoretical importance. This paper studies the quantitative asymptotic stability of the optimal filter for signal-observation models with linear drift and additive Brownian noise. When the initial law is additionally assumed to be Gaussian, the optimal filter reduces to the celebrated Kalman-Bucy filter, for which the detectability condition on the signal and observation matrices is the sharp condition ensuring asymptotic stability. In the linear setting where the detectability condition holds but the initial law need not be Gaussian, we make the following contributions by studying the Zakai equation satisfied by the unnormalized filter. First, we study the asymptotic behavior of the deterministic part of the Zakai equation in the general case with nonlinear drift, and place the detectability condition in this context. Second, in the case of linear drift, we characterize the solution of the Zakai equation (hence the optimal filter) by a gauge transformation that separates the finite-dimensional stochastic dynamics from the infinite-dimensional deterministic dynamics. Third, we use this representation to establish asymptotic exponential filter stability in the $\chi^p$-divergence (with respect to a reference optimal filter), the total variation norm, and the $2$-Wasserstein distance.

The Beardwood-Halton-Hammersley theorem characterizes the asymptotic length of the optimal Euclidean traveling salesman tour through independent and uniformly distributed random points $X_1,\ldots,X_n$ in the unit square. It states that there exists a universal constant $\beta$ such that the length of the shortest tour is asymptotic to $\beta \sqrt{n}$ almost surely. The best bounds established to date are $0.6277\leq \beta \leq 0.90367$. In this paper, we improve these bounds to $0.6421\leq\beta\leq0.8810$. Using importance sampling, we further show that $0.6536\leq \beta\leq 0.8749$ holds with probability at least $1-2\times 10^{-4}$. Here the probability is taken with respect to the randomness of the sampling procedure.

In this paper, we study favorite sites of one-dimensional transient random walk on $\mathbb{Z}$ beyond the simple case. Denote by $\mathcal{K}(n)$ the set of favorite sites at time $n$. The goal of this paper is twofold. First, we study $\# \mathcal{K}(n)$, the cardinality of $\mathcal{K}(n)$, and show that, under some mild conditions, for any positive integer $k$, $\mathbf{P}(\#\mathcal{K}(n)=k, {\rm i.o.})=1$, and $\limsup_{n\to\infty}\#\mathcal{K}(n)/\log\log n=-1/\log \gamma$, where $\gamma$ is the probability that the random walk never returns to the starting point. Second, we study the escape rate of favorite sites, and give the limsup and liminf behaviors, where the liminf behavior is characterized via an integral test. To the best of our knowledge, this integral test is the first result of this nature for random walks (in non-random environments).

Rates of convergence in normal approximation are fundamental to probability and statistics. The theory has evolved from normalized sums to Studentized statistics, smooth functions of sample means, and $U$-statistics, and more generally to symmetric statistics. A central question throughout this development has been to identify the weakest assumptions under which optimal rates of convergence can be established. First, in this paper we address this question by establishing optimal bounds for adjusted normal approximations of symmetric statistics based on independent and identically distributed observations. More precisely, we bound the Kolmogorov error by a very neat sum of the classical third-moment term and the variance of the remainder term. This resolves a long-standing conjecture posed by Bentkus, Jing and Zhou (Ann. Probab. 37 (2009), 2174–2199). Secondly, we establish the optimal Berry-Esseen rate for jackknife Studentized \(U\)-statistics of order two under minimal conditions, i.e., a finite third absolute moment of the first projection and a finite \(5/3\)-absolute moment of the canonical kernel. This resolves a long-standing conjecture in classical probability theory. Our proofs combine signed Prawitz smoothing with a conditional-variance submartingale, polynomial approximation, and a bounded change of measure.

In this article, we construct, for a broad class of one-dimensional stochastic reaction-diffusion equations, a sequence of spatially-structured individual-based models whose rescaled empirical processes converge to the solution of the equation. The purpose of this article is to show that, whenever the rescaled one-step mean and variance of a reasonable individual-based model converge to those of an SDE with drift $b(x)x(1-x)+\theta^+(1-x)-\theta^-x$ and noise coefficient $\sqrt{\sigma(x)x(1-x)}$, under certain moment conditions there also exists a corresponding multi-island model, composed of independent and identically distributed copies of the individual-based model on each island, with selection and mutation scaled to the diffusive time scale, coupled through migration, which converges to an SPDE with a Laplacian plus the same drift and noise coefficients, provided this SPDE is unique in law.

We study two interacting reinforced occupation processes driven by a common two-state Markov environment whose switching probabilities decrease at the same rate as the stochastic-approximation gain. At this critical scale, the environmental motion persists in the first-order limit, which is a telegraph-driven piecewise deterministic Markov process. Consequently, centering at a deterministic equilibrium does not separate the order-one environmental response from the intrinsic square-root fluctuations. To overcome this obstruction, we introduce an exact discrete random centering that removes the accumulated environmental forcing before rescaling. An exact decomposition into collective and synchronization coordinates then separates the two contraction mechanisms of the system. Under the contraction condition corresponding to the square-root regime, we prove joint functional convergence of the randomly centered coordinates and the first-order background to a stationary hybrid PDMP–diffusion system. Its stationary entrance law is characterized through stochastic convolutions over the remote logarithmic past. Conditionally on the complete environmental path, the two fluctuation modes are independent centered Gaussian processes with a common state-dependent volatility and different contraction rates. We also determine the invariant distribution of the first-order component, derive explicit stationary second moments, and obtain a closed-form correlation formula that recovers interaction information not visible in the synchronized first-order limit.

For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $\Theta(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $\Theta_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.

We derive a weighted Kirchhoff diffusion on a smooth Riemannian open book as a scaling limit of regime-switching Brownian motions on boundaryless manifolds. The pages are compact Riemannian manifolds sharing a common boundary manifold and inducing the same metric on the binding, while their second fundamental forms and interior geometries may differ. For each $\varepsilon>0$, we regularize the metric in an $o(\varepsilon)$ collar, pass to smooth doubles of the pages, run Brownian motion on each double, and allow the page label to switch in a collar of width $\varepsilon$ at rates \(q_\varepsilon(r)Q_{ij},\) where $Q$ is an arbitrary irreducible finite-state Markov generator with invariant law $\pi$, while \(q_\eps\) is bounded and supported on an \(\eps\) collar. Denoting \(\Theta_\eps\) the total mass of \(q_\eps\) and \(\Xi_\eps\) its second raw moment, we prove that, if \(\Theta_\varepsilon\to\infty\) and \(\Xi_\varepsilon\to0\), then the glued processes converge, from every deterministic sequence of starting points, to Brownian motion with weigthed Kirchhoff interface conditions. No reversibility of $Q$, pointwise scaling ansatz for $q_\varepsilon$, or first-moment condition is required. Furthermore, we show that the second fundamental form of the pages contributes no additional interface term: curvature remains only through the bulk Laplace–Beltrami operators.

We study rumor-spreading models on the one-dimensional integer lattice, with particular emphasis on the effect of forgetting mechanisms on the long-term behavior of the rumor. We consider the spatial version of the classical Maki–Thompson model together with two variants that incorporate forgetting. Each model is formulated as an interacting particle system and describes the propagation of information through local interactions between neighboring individuals. Our results show that the introduction of forgetting fundamentally changes the behavior of the process. In the classical Maki–Thompson model, the rumor becomes extinct, whereas the two models with forgetting exhibit nontrivial phase transitions between extinction and survival. More precisely, we establish the existence of critical transmission rates that separate regimes in which the rumor dies out almost surely from regimes in which it survives with positive probability. We also investigate the dependence of these critical thresholds on the stifling rate. Combining rigorous arguments with numerical simulations, we prove the existence of the phase transitions, obtain estimates for the critical values, and explore the corresponding phase diagrams. These findings show that forgetting, despite acting as a mechanism that removes spreaders, can qualitatively alter the dynamics of rumor propagation by creating regimes of sustained information spread. Our work provides new insight into the critical behavior of spatial rumor models and raises several questions for future research.

We prove a large deviation principle for the area of a two-pointed weight-$W$ quantum disk conditioned to have left boundary length one, where $0<W<2$ is fixed and $\gamma\downarrow0$ without a subsequence restriction. Conformal welding and the one-block mating-of-trees law give independent Brownian blocks and a recursive area decomposition. Growing moments and negative Laplace transforms lead to two Hamilton–Jacobi equations, whose characteristic solutions determine the limiting transforms. Direct comparison proves moment convergence; compact-interval comparison and a change of measure prove Laplace convergence. The resulting good rate function is specified by scalar endpoint equations and is elementary when $W=1$.

Fix $W\in(0,2)$ and let $\gamma\downarrow0$. We prove that a weight-$W$ quantum disk conditioned to have unit quantum area converges, in the maximum-embedded strip, to the deterministic field $\phi_W(t+i\theta)=\log(W/(4\pi))-2\log\cosh(Wt/2)$ and its constant-curvature area measure. The proof establishes a joint large-deviation principle for the radial Brownian input, the lateral Gaussian free field, and the total Gaussian multiplicative chaos mass. Diverging-order moment bounds justify both adjoining the noncontinuous area functional and retaining its unbounded positive power in the unit-area expectation. An exact radial Girsanov shift centers the candidate, and reflection together with the sharp Onofri inequality proves its unique optimality. This yields concentration without a quadratic expansion of the running maximum or an order-one partition-function asymptotic. We also identify the field rate with the renormalized Liouville action in the maximum embedding.

Fractional Brownian motion provides a useful framework for modeling random phenomena with anti-persistent temporal correlations that arise in applications such as anomalous diffusion, hydrology, finance, and geophysical processes. In this paper, we study a class of semilinear stochastic partial differential equations driven by additive fractional Brownian motion with Hurst parameter $H\in(0,\frac12)$. We establish the well-posedness and space-time regularity of the mild solution and investigate the strong convergence of a stochastic exponential integrator for the temporal discretization. The analysis allows the linear operator to be non-self-adjoint and combines analytic semigroup estimates with the canonical Hilbert-space structure associated with fractional Brownian motion and Malliavin calculus to handle the low temporal regularity of the noise. We derive an $H$-dependent strong temporal convergence rate, which reaches $H+\frac12$ under maximal spatial regularity. Numerical experiments for a stochastic advection–diffusion–reaction problem with heterogeneous Darcy flow confirm the theoretical temporal convergence rates. We also estimate the mean of the solution with different Hurst parameters $H\in(0,\frac12)$ using the Monte Carlo method.

Let $P$ be a Markov operator on a general state space $(S,\Sigma)$ with invariant probability $m$, assumed ergodic. When $P$ is normal in $L_2(m)$, we prove the necessity of uniform ergodicity, under some spectral conditions (satisfied when $P$ is symmetric), for the annealed CLT to hold for every centered $ f \in L_2(m)$. We also obtain results without assuming normality under some Ritt-type conditions.

Choose a uniformly random ordering of the vertices of a finite tree and run the usual greedy maximal-independent-set algorithm. We compare the resulting law on maximal independent sets with the uniform law. We prove that exact uniformity occurs only for the one-vertex tree and the single edge. The proof is structural: a diameter endpoint exposes a pendant star, and the remaining one-pendant-leaf case is resolved by a strict injection between exact permutation fibres obtained by swapping the pendant leaf with its support vertex. Exact uniformity is therefore rigid, but it can be approached closely. For an explicit mixed-spider family $T_{k,l}$ we count the maximal independent sets and compute the exact probability of every output. With $l=2^k-k$ the total-variation bias is positive and satisfies [ b(T_{k,2^k-k})=O!\left(\frac{\sqrt{k}}{4^k}\right) =O!\left(\frac{\sqrt{\log n_k}}{n_k^2}\right), \qquad n_k=2^k+k+1. ] The theorem package has also been formalised in Lean and registered with Palomar. These records document machine-checked formal verification and the checked axiom boundary; they are not peer review or a certificate of novelty.

We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $\Omega(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

We consider the problem of sampling from Gibbs distributions on matrix spaces whose potential energies are neither convex nor globally gradient-Lipschitz. We introduce a family of non-quadratic kinetic energies that lead to a new underdamped Langevin system with momentum preconditioning, in which the gradient of the kinetic energy acts as a smooth spectral taming of the momentum. We prove that, under these relaxed assumptions on the potential, the resulting dynamics leaves the target Gibbs measure invariant, and we establish exponential convergence to equilibrium in a weighted total variation distance. Finally, we show that the corresponding Euler-Maruyama discretization admits moment bounds that are uniform in time, without any modification of the potential gradient, which ensures the stability of the resulting sampling algorithm.

Motivated by applications in random geometry, we investigate the notion of subordination of snakes in the discrete setup. More precisely, given a random walk $W$ indexed by a tree $T$ and with steps in $\{...,-1,0,1\}$, we consider its subordinate tree obtained by contracting every edge of $T$ that does not lead to a new record of the walk $W$. When the underlying tree $T$ is a Bienaymé-Galton-Watson tree, we characterize the distribution of this subordinate tree. In particular, when $T$ has a critical offspring distribution in an $\alpha$-stable domain of attraction with $\alpha\in(1,2]$, and under a light tails assumption on the steps, we prove that the associated subordinate tree is itself a Bienaymé-Galton-Watson tree with an offspring distribution in an $\frac{\alpha+1}{2}$-stable domain of attraction. Along the way, we obtain the asymptotic tail of the maximal displacement of the critical branching random walk $W$ in this stable regime, under minimal assumptions. Finally, we use these results to prove scaling limit statements about the subordinate tree.

We give sufficient conditions ensuring that, at every fixed positive time, the scalar backward component of a Markovian forward-backward stochastic differential equation with multidimensional forward process admits a density with respect to Lebesgue measure. Existing density criteria for BSDEs often obtain Malliavin non-degeneracy through sign or monotonicity assumptions. We develop a different route for a scalar backward component with an arbitrary-dimensional forward state. Under regularity assumptions and a structural compatibility condition on the generator, the terminal condition is only required to be non-constant. The key idea is to deduce Malliavin non-degeneracy from deterministic rigidity of the critical set of the decoupling field. We combine Malliavin calculus with unique continuation and backward uniqueness for the associated semilinear parabolic equation. Unique continuation precludes the spatial gradient of the decoupling field from vanishing on a set of positive measure unless it vanishes identically on that time slice, while backward uniqueness propagates such vanishing to the terminal time. Along the way, we establish a unique continuation property from sets of positive measure and a backward uniqueness result on the whole space for the linear parabolic systems arising from differentiated semilinear equations.

There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools – spectral analysis and mean first-passage time (MFPT) analysis – and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.

We study the transverse field Ising model, defined by the Hamiltonian $H =\frac{1}{2}\sum_{i, j\in [n]} J_{ij} Z_i Z_j +\sum_{i=1}^n h_i^z Z_i + \eta\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $\eta$ is the transverse field strength. Let $\Delta(J)=\lambda_{\max}(J)-\lambda_{\min}(J)$ be the spectral width of $J.$ When the inverse temperature $\beta\geq0$ satisfies $\Delta(J)\cdot\frac{\tanh(\beta\eta)}{\eta}\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(\beta)=\operatorname{Tr}(e^{-\beta H})$ to a given relative error $\epsilon\in(0,1)$ in time polynomial in $n$, $\beta$, the model parameters, and $\epsilon^{-1}$. When $ \Delta(J) \cdot \frac{\tanh(\beta \eta)}{\eta} > 1 ,$ we show that approximating $ Z(\beta)$ within an $\exp(o(n))$-multiplicative factor is $\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $\Delta (J)\cdot \frac{\tanh(\beta \eta)}{\eta}\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ \rho_\beta = \frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \in \{I, X\}^{\otimes n}$, the algorithm further achieves arbitrarily small relative error.

We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$, where the support of the optimal coupling is sparse. For smooth marginal densities in $\mathbb{R}^d$, we show that the support contains the graph of the Brenier map. After centering by the Brenier image and rescaling by $\ell=\varepsilon^{1/(d+2)}$, the sections of the support have explicit ellipsoidal limits for $\varepsilon\to0$, except at boundary points, where the limits are half-space profiles. The optimal dual potentials admit expansions at order $\ell^2$, uniformly up to the boundary. We identify the leading coefficients, which consist of a common local profile and opposite global corrections determined by a linear Neumann problem. Finally, we analyze two approximations to the Brenier map, namely the gradient of the dual potential and the conditional mean of the coupling. We obtain the sharp $L^p$ rate $\ell^{1+1/p}$ with exact leading constants and further identify the leading interior and boundary biases. Taken together, our results illustrate that quadratic regularization induces an accurate sparse approximation of classical optimal transport.

The critical Ising model admits several natural scaling limits. For an Ising model on a lattice approximation $D^a$ of a planar domain $D$, the rescaled spin field converges, as $a\to0$, to the critical Ising magnetization field (IMF), a rough random distribution on $D$, while the spin interfaces converge to a nested $CLE_3$ and the FK–Ising representation converges to a colored $CLE_{16/3}$. In this work, we prove that the IMF can be directly reconstructed from the nested $CLE_3$. This complements the previously known construction of the IMF from the colored $CLE_{16/3}$ [CGN15]. Our result also implies that %We further show that if one constructs a colored $CLE_{16/3}$ from the nested $CLE_3$ through CLE percolation [MSW17], the resulting construction recovers the same magnetization field. A main difficulty in this present reconstruction comes from the fact that the renormalization exponent of the IMF, $15/8$, is smaller than the Hausdorff dimension of the $CLE_3$ carpet, $187/96$. Consequently, the field cannot be recovered by a direct Minkowski-content construction from the nested $CLE_3$ as is the case for $CLE_{16/3}$. We conjecture that the converse measurability also holds, namely that the nested $CLE_3$ can itself be reconstructed from the IMF.

Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $\Gamma$-independent semicircular elements. This characterizes their limiting spectral distribution and yields improved bounds on their spectral norm, a key quantity in the analysis of algorithms for Tensor PCA. Finally, we show that, in an appropriate double limit, independent Kikuchi matrices converge to the $q$-Gaussian system, another central object in noncommutative probability.

We develop a method for proving matrix concentration inequalities by identifying random matrices with associated deterministic operators acting on suitable Fock spaces and controlling norms of these operators and their restrictions to low-order subspaces. Applying this method, we obtain new strengthenings of the non-commutative Khintchine inequality that improve on the state of the art, in particular sharpening recent inequalities due to Bandeira, Boedihardjo, and van Handel (2023) quantifying intrinsic freeness of random matrices. Our proofs of these results are based on relatively simple operator algebra arguments and involve neither Gaussian interpolation nor explicit combinatorics of tracial moments. Further, our techniques apply equally well to several models of non-commutative random variables of interest in the literature, such as operator series constructed from $q$-Gaussian and $\Gamma$-independent systems of operators, treating all of these objects with the same method. We obtain new norm bounds for such operators both in the style of the non-commutative Khintchine inequality and in the style of Lehner's operator norm formula.

We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on $(\mathbb Z/N\mathbb Z)^2$, let $T_N$ denote its cover time. We prove that $\frac{T_N}{(2/\pi)N^2\log N}-2\log N+\log\log N \Longrightarrow G+\log(\kappa Z)$, where $G$ is a standard Gumbel random variable, $Z$ is the total mass of the critical Gaussian multiplicative chaos associated with the zero-average Gaussian free field on the unit torus, $G$ and $Z$ are independent, and $\kappa>0$ is deterministic. This answers the limit-law question suggested by Aldous and recorded by Dembo, Peres, Rosen and Zeitouni. The proof identifies the random fluctuations in the number of small, well-separated unvisited components at a deterministic time before coverage, and then estimates the time needed to visit the remaining components.

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic \(h\)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

We study continuous-time Galton–Watson trees whose offspring generating function takes the form \begin{align} \label{eq:gen} f(s) = s - (1-s)^\alpha L(1-s), \end{align} where $\alpha \in (0,1)$ and where $L:[0,1] \to [0,\infty)$ is slowly varying at zero. Processes with this offspring generating function explode in finite time. We observe that, conditional on explosion at time $T$, at each earlier time $t < T$ there is a unique particle $\xi_t$ alive at time $t$ who is an ancestor of all but finitely many particles at the explosion time. We call $(\xi_t)_{t \in [0,T)}$ the spine to explosion, and show that the births off the spine admit a Poissonian description, where they become both more frequent and larger as $t \uparrow T$. We undertake a careful study of the size of the population leading up to explosion, showing in particular that near explosion, the rescaled population is approximately gamma distributed. More generally, conditional on explosion at time $T$, define a stochastic process $Z^\varepsilon := (Z_t^\varepsilon)_{t \in \mathbb{R}}$ by setting \begin{align*} Z_t^\varepsilon := E(\varepsilon e^{-t})N_{T-\varepsilon e^{-t}}, \qquad E(t) = \mathbf{P}(N_t = \infty). \end{align*} We show that as $\varepsilon \downarrow 0$, $Z^\varepsilon$ converges in finite-dimensional distributions to a stationary continuous-state branching process with immigration whose branching mechanism is subcritical and whose immigration mechanism corresponds to spine events. Finally, we show that this limiting process has a Markovian time reversal, and that the coalescent process associated with it is a stochastically time-changed Beta$(2-\alpha,\alpha)$-coalescent.

Consider the circular beta-ensembles, Jiang and Matsumoto computed the moments of traces, and proved central limit theorems for polynomial linear statistics using Jack functions. We give a direct proof of the Gaussian limits for the traces of powers of circular beta-ensembles using the theory of orthogonal polynomials on the unit circle and Stein's method. The proof relies on an explicit coupling of the Verblunsky coefficients of circular beta-ensembles, which also allow us to obtain almost sure and $L^2$ convergence.

In a recent paper [Kwaśnicki, Poisson kernels on the half-plane are bell-shaped, Bull. London Math. Soc. 58 (2026), no. e70303], the author proved that Poisson kernels for a class of elliptic operators in the half-plane are bell-shaped. By allowing one-sided barriers, we prove a converse result.