Papers

We derive the half-space KPZ equation as a continuum limit from a general class of Ginzburg-Landau SPDEs under weak nonlinearity scaling. To our knowledge, this is the first half-space universality result for the KPZ equation beyond integrable models (and their perturbations).

We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $\beta_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=\alpha k$ with fixed $\alpha>1$, both Prange's algorithm and DQI can sample at any inverse temperature $\beta < \tanh^{-1}(1/\alpha)$ for sufficiently large $k$, well beyond the dynamical threshold $\beta_{\rm dyn} \sim \sqrt{2\ln k / (\alpha k)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.

We develop a spectral theory for dynamical large deviations in non-Markov jump processes and non-Markov chains, whose dynamics depends on the past through state- and jump-dependent empirical observables. We demonstrate that a multiscale Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) Ansatz separates fast configurational relaxation from slow memory evolution, reducing the Feynman–Kac equation for occupation and flux statistics to an eigenvalue problem for a new tilted operator coupled to Hamilton–Jacobi characteristics. This provides a computationally efficient framework for quantifying fluctuations in a broad class of non-Markovian systems. We illustrate our general results with a bistable self-induced East model.

Protocols for quantum state tomography can be characterized either by their sample complexity or by the tomographic rate function that governs the large deviation behaviour of error estimates. We establish a framework for the analysis of covariant protocols which allows us to compare these measures. First, we derive the rate functions for several protocols, including random purification-based protocols and the sample-optimal protocol of Haah et al. These rate functions are expressed in terms of a new family of divergences which includes the reverse sandwiched Rényi divergence and Keyl's annealed quantum relative entropy as special cases. We show that these rate functions obey a strict ordering, even though the sample complexity of each protocol is order optimal. We consider a different version of sample complexity based on Wasserstein distance rather than trace distance, and show that the ordering of the rate functions implies analogous strict inequalities for the Wasserstein-type sample complexity. Thus the tomographic rate function is a more fine-grained indicator of the performance of a tomography protocol than the usual sample complexity.

We study the distribution of the zeros of $\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely weakly converges as $N\to\infty$ to a deterministic measure, depending on just the variance and covariance of the entries of the $A_j$. This generalizes the circular, elliptic, and semicircle laws, which concern the special case of this question where $d=1$. Unlike those classical laws, these measures can combine nonuniform two-dimensional densities with singular components supported on curves, producing a variety of petal-shaped regions, inspiring our name “the petaloid law”. We give explicit formulas for the limiting densities and supports. Under a Gaussianity assumption, we also show that there are almost surely no eigenvalues outside small neighborhoods of the limiting support.

We develop a mixing time method for estimating the sample complexity of quantum state discrimination. We start with considering the minimum-error discrimination of geometrically uniform pure state ensembles, and prove that its sample complexity has a tight estimate given by a quantum homogeneous mixing time [George et al., 2026] and a quantum version of the generalized Dobrushin coefficient [Wolfer, 2020]. This quantum mixing time further reduces to a classical one when the generating group $G$ forms a Gelfand pair with the stabilizer subgroup $H$ of the generator state. In this case the generalized Dobrushin coefficient can be fully expressed by representation-theoretic quantities of the commutative Hecke algebra $\operatorname{End}_G(\mathbb C[G/H])$. In particular, this method reduces the sample complexity estimation of learning quantum coupon collector states [Arunachalam et al., 2020] and learning phase states to classical mixing time problems. We apply this framework to answer the open problems of learning degree-$d$ phase states over $\mathbb F_q$ in [Alrabiah et al., 2026] and generalized Boolean phase states over $\mathbb Z_q$ [Arunachalam et al., 2023]. The framework also applies to hypergraph state ensembles, giving estimates expressed fully in terms of hypergraph data and recovering estimates for graph state ensembles in [Montanaro and Shao, 2022]. Finally, we extend the discussion to arbitrary mixed state ensembles with uniform priors, prove a sandwiched bound for minimum-error discrimination sample complexity by a quantum weakly mixing time, and provide a tight estimate for the minimax discrimination sample complexity from [D'Ariano et al., 2005] by a Dobrushin-type coefficient. We also discuss the method of strengthened data processing inequality [Gao and Rouzé, 2022] and give an upper bound in terms of a strengthened data processing inequality constant.

We determine which representation sectors control the slowest convergence of random Pauli rotations and several local random quantum circuits. For $d=2^n$, the continuous Pauli walk has exact all-representation gaps $$\Delta_{\mathsf{SU}(d)}=\frac{d-8}{8(d-1)}\quad(n\ge4),\qquad \Delta_{\mathsf{PU}(d)}=\frac{d(d-3)}{8(d^2-1)}\quad(n\ge3).$$ The special-unitary gap is attained in $\bigwedge^8\mathbb{C}^d$ and is invisible to balanced moments, while the projective gap is already attained in a balanced fourth-moment representation. We obtain exact discrete and orthogonal Pauli analogues, exact finite-Clifford chain/path gaps, and dimension-independent all-representation Haar brickwork lower bounds; on three qubits every weighted Haar-$\mathsf{SU}(4)$ edge walk reduces to one universal principal-angle constant. For real complete-graph models, the entire ordinary fourth-tensor gap reduces to a quadratic-size occupation operator, is simple for every $n\ge4$, and has an expansion beginning $$\Delta_n^{(4)}=\frac{10}{9n}-\frac{88}{105n^2}+O(n^{-3}),$$ with competing branches separated only beyond all algebraic orders. For the two finite real Clifford complete-graph walks, this fourth-tensor mode is the exact all-representation group gap for every $n\ge144$; in the same range it simultaneously governs every ordinary even tensor degree $k\ge4$. Thus different representation scopes can have genuinely different slow modes.

For each $n \in \mathbf{N}$, let $T_n$ be a uniformly random (rooted, Type I) triangulation of the sphere with $n$ vertices, viewed as a metric space equipped with its graph distance. We show that for every $\delta>0$, with probability tending to $1$ as $n \to \infty$, every embedding of $T_n$ into a separable Hilbert space has distortion at least $(\log n)^{1/4-\delta}$.

We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies $H_2(\mu)=-\log\|\mu\|_2^2>12$, then $\mathbb{P}(\operatorname{disc}(R)\text{ is a square})=O_\mu(1/\log n)$. Combined with previous results, this shows that, for such measures $\mu$ and under additional hypotheses, $R$ has full Galois group $\mathrm{Sym}(n)$ with high probability, when conditioned on $\omega_0\ne 0$.

We study discrete-time quantum trajectories of a finite-dimensional system in a disordered environment, where the instrument applied at each step is determined by an invertible measure-preserving dynamical system. For general quantum instruments on a standard Borel outcome space, we construct the quenched probability law on sequences of measurement outcomes for a fixed realization of the disorder and a given initial state. Under a quenched Doeblin condition with a uniform deterministic minorization constant and an environment-dependent minorizing probability measure, we establish the existence and uniqueness of an equivariant family of random posterior laws and identify its barycenter as the unique dynamically stationary state for the associated non-selective channel cocycle. If the environment is ergodic, we prove a pathwise ergodic theorem valid for every measurable choice of initial state: for almost every realization of the disorder, time averages of posterior states converge almost surely under the corresponding quenched law to the disorder average of the unique dynamically stationary state. For an ergodic environment, we establish quenched variance asymptotics and, when the asymptotic variance is positive, a central limit theorem and standardized Berry–Esseen bounds for additive functionals generated by bounded measurable real-valued functions of the environment and posterior state. These results hold for every measurable initial state and almost every disorder realization. We then show that the stationary annealed joint process of instruments and posterior states inherits the $\alpha$-mixing of the instrument process, up to an exponentially decaying term. Finally, we provide classes of examples involving perfect and imperfect measurements, including models with continuous or finite outcome spaces, that satisfy the standing Doeblin condition.

We study irrevocable maximum-cardinality matching in trees revealed by successive leaf attachments, with a known horizon and an exogenous growth law that is misspecified or unknown. For deterministic affine attachment forecasts with nonnegative degree reinforcement, the optimal threshold policy loses at most twice the cumulative expected conditional total-variation error relative to an online oracle knowing the actual growth law. This follows from a unit-span property of the Bellman continuation score and has no additional horizon factor. A four-vertex example attains the coefficient two for the specified deterministic policy, and a two-model argument gives a lower bound linear in the model-error budget for arbitrary policies under general misspecification. For uniform-preferential attachment, the local error has an exact expression through the leaf count. When its constant mixture parameter is unknown, we estimate it from the same growing tree and update the threshold policy at geometric times. A parameter-sensitivity bound for individual Bellman prices and uniform degree-moment estimates yield expected regret $O(\sqrt{n}\log^2 n)$, using $O(n^2\log n)$ arithmetic operations and $O(n)$ stored entries. The exact minimax rate remains open.

In the adjacent transposition shuffle on a cycle, $n$ distinct cards are placed at the vertices of a cycle, and adjacent cards swap positions according to independent Poisson clocks of rate 1. We prove that this Markov chain exhibits total variation cutoff at time $n^2\log n/(8\pi^2)$, with a window of order at most $n^2\log\log n$.

We study the problem of optimally placing a large number of small absorbing traps to minimize the mean first-passage time (MFPT) of particles diffusing in a heterogeneous medium with space-dependent diffusivity $D(x)$ and prescribed initial particle distribution $\omega(x)$. In two and three dimensions we identify two distinct regimes, depending on how strongly the trap ensemble depletes the particles. In the weak trapping regime (fewer traps) the optimal trap density $\rho$ is the arithmetic average $\rho = \frac12(\mu+\omega)$ of the normalized inverse diffusivity $\mu \propto 1/D$ and the initial particle distribution $\omega$. In the strong trapping regime (more traps), it is proportional to their geometric average, $\rho \propto \sqrt{\mu \omega}$. We revisit the classical Green's function approach and show that it applies to weak trapping only; in 2D this requires the trap size to be exponentially small in the number of traps, so that most applications of interest fall in the strong trapping regime instead. We therefore develop a homogenization approach that captures both regimes. A special choice of initial particle distribution is $\omega \propto 1/D$, which corresponds to the equilibrium particle distribution in the absence of traps according to the Itô interpretation. In this case we show that the optimal trap distribution is the stationary particle distribution itself, regardless of the trapping strength, and this makes the MFPT constant throughout the domain; the same trap distribution also minimizes the worst-case capture time. Finally, in one dimension we find instead $\rho\propto\left(\mu \omega\right) ^{1/3}$, and a similar analysis covers space-dependent drift and thin domains of variable cross-section. Direct numerical optimization confirms the analytical results.

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

We introduce a simple reaction-network model of a quorum-sensing population that couples the cell density $x$ to the signal density $w$. In the four-channel cell-signal network studied here, scaling signal production and removal by the same factor $r$ leaves all deterministic equilibria, and their stability types, unchanged. Nevertheless, we show that $r$ shifts the quasipotential barrier $\Delta V(r)$ for rare transitions towards extinction, and thus, under metastable exit assumptions, the mean time to reach a fixed neighbourhood of the extinction state on the exponential scale $e^{N\Delta V(r)}$. We compute the barrier by minimization of the path action with the signal retained as a fluctuating coordinate, and we compare it with exact stochastic simulation of population-threshold crossing times regressed in $N$. Over a range of $r$, the minimum-action barriers satisfy $\Delta V(r)=\Delta V_\infty+O(1/r)$, where $\Delta V_\infty$ is obtained by eliminating the signal first. At $r=1$, the barrier is 55% larger than $\Delta V_\infty$. The saddle barrier $\Delta V(r)$ is also the least action needed to enter the basin of extinction, but the density threshold can be crossed more cheaply: at $r=0.5$ the cheapest crossing costs 7.7% less action, keeps the signal high, and is usually followed by recovery. Simulated arrival times in this neighbourhood, which include failed attempts, grow with slopes within two fitted standard errors of the saddle barrier at every tested rate, and within $0.004$ of it if the logarithmic prefactor term is omitted. Under either regression model they exclude $\Delta V_\infty$ at $r\le2$ by at least $4.4$ fitted standard errors, without using the action solver.

We prove a continuous-time Carleson embedding theorem with constant $e$ for a system of square-integrable continuous martingales whose quadratic covariations mimic the generalised Cauchy–Riemann relations. The argument is based on a Bellman function and a multidimensional Itô formula. As an application we transfer the estimate to the upper half-space via the Gundy–Varopoulos representation and obtain a Carleson embedding, still with constant $e$, for the vector consisting of a function and its Riesz transforms in arbitrary dimension.

We study the width required for a randomly initialized hidden layer of a neural network to achieve rank lifting. Namely, given a dataset $X \in \mathbb{R}^{m \times d}$ of $m$, $d$-dimensional input vectors separated by an angle of at least $\theta$, we consider the random feature matrix $\sigma(XR)$, where $R$ is standard Gaussian. For positively homogeneous nonpolynomial activations, which include sign, Heaviside, ReLU, and ReLU powers among others, we prove that $$n \gtrsim \frac{1}{\theta}\max\left\{m,\log\left(\frac{1}{\delta}\right)\right\}$$ neurons suffice for $\sigma(XR)$ to have full row rank $m$ with probability at least $1-\delta$. This dimension-free bound exponentially improves the previous general-dimensional guarantee for sign features (Drago et al., 2026) and is essentially tight. The proof shows that one random feature column escapes every proper subspace of $\mathbb{R}^m$ with probability $\Omega(\theta)$, using a coupling of nearby Gaussian directions and a local crossing of the induced hyperplane arrangement. We also study stable rank lifting, where the goal is to establish a quantitative analogue of exact rank lifting, i.e., a lower bound on the smallest eigenvalue of the empirical feature Gram matrix in high-probability. Our analysis unifies and generalizes stable rank guarantees for all $q$-homogeneous non-polynomial activations following prior work in Panigrahi et al. (2020) and Song (2026). In particular, we combine a diagonally dominant Taylor tail of the population kernel with truncation and matrix concentration, to show that for positively homogeneous nonpolynomial activations, stable rank lifting is achieved at width $$n \gtrsim C^q \frac{m}{\theta^{2q+1}} \log^{2q+\frac{1}{2}}\left(\frac{m}{\theta}\right) \log\left(\frac{m}{\delta}\right),$$ where $q$ is the degree of the activation and $C > 0$ is some universal constant.

Landmark shape spaces arise from configurations of $n$ distinct landmarks with rigid motions factored out and scale normalized. When equipped with sufficiently regular metrics inherited from right-invariant metrics on diffeomorphism groups, the shape spaces inherit geodesic completeness from geodesics of diffeomorphisms. We now answer the corresponding stochastic question if the Riemannian Brownian motion on landmark shape spaces exists for all time and hence rule out initially distinct landmarks colliding when following a Brownian flow. We show that with general classes of metrics, including degenerate metrics with infinitesimal rigid motions in their null space, landmark shape spaces are stochastically complete, thus making the use of Riemannian Brownian motion for modelling shape stochasticity in applied fields well-founded.

We establish nonasymptotic spectral comparison results for matrices whose entries are non-linear functions of an iid random field. Under an exponential decay assumption on an $\mathbb L^\infty$-coupling coefficient, we derive high-probability bounds for the Hausdorff distance between their spectra and those of Gaussian matrices with matching covariance structures. We further derive a comparison with a covariance-matched free model and derive corresponding noncommutative Khintchine-type bounds. The results apply, in particular, to matrices whose entries are nonlinear transformations of causal linear processes, Volterra-type processes, and neural networks. The proof combines finite-memory approximation, block decomposition, Gaussian interpolation, and resolvent estimates.

We study large deviations for rare clusters in sparse systems of moving particles. In the regime \(nr_n^d\to0\) and \(\rho_{k,n}=n^kr_n^{d(k-1)}\to\infty\), we prove a large deviation principle for the empirical measure of isolated \(k\)-particle trajectory clusters. The speed is \(\rho_{k,n}\), and the rate function is the relative entropy \(h(\,\cdot\,\mid\tau_k)\), where the finite reference measure \(\tau_k\) explicitly incorporates the underlying path law. As consequences, we derive free-energy variational formulas for bounded interactions and a hard-core constraint and identify the corresponding minimizing cluster law. The normalized optimizer provides the basis for Metropolis–Hastings sampling of interacting trajectory clusters. As an application motivated by chain formation and swarming in active particle systems, we calibrate the resulting stochastic model to experimental trajectories of magnetic micromotors and find that the fitted velocity scale varies systematically with particle size and magnetic forcing.

Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\ast$-subalgebras of the complex matrix algebra ${\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every pair of elements, one from each subalgebra. Writing such a pair as ${\sf D}_n$ and $U{\sf D}_nU^*$ with $U$ unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition $|\det U[I,J]|^2=\binom{n}{r}^{-1}$ for every $1\le r\le n$ and all $I,J$ with $|I|=|J|=r$. We show that this condition holds if and only if $n\le3$ and $\sqrt{n} U$ is a complex Hadamard matrix. To quantify the failure of exact realization for $n\ge 4$, we introduce the uniform-minor discrepancy $\delta_r(U)$. We identify it with the mean-square error in the $r$-th coefficient of finite free multiplicative convolution for two diagonal matrices whose diagonal entries are independent and uniformly distributed on the unit circle. We establish the symmetry $\delta_r(U)=\delta_{n-r}(U)$ and the monotonicity $\delta_1(U)\le\delta_2(U)\le\cdots\le \delta_{\lfloor n/2\rfloor}(U)$. For flat unitaries, we derive an explicit formula for $\delta_2$, yielding $\delta_r(U)\ge\frac{n-3}{2n}$ for $2\le r\le n-2$. Equality for $r=2$ holds precisely when the entrywise square of $\sqrt{n}U$ is also complex Hadamard.

For any $t\in [0,1/2)$, consider the process $\eta_t:[0,1/2]\mapsto\mathbb{R}$ defined by $\eta_t(s)=\Pi(t+s)-\Pi(t)$, where $\Pi$ is the directed geodesic from $(0,0)$ to $(0,1)$ in the directed landscape. Let $0\leq t<u<1/2$. We show that the laws of $\eta_t$ and $\eta_u$ are mutually absolutely continuous if and only if $t>0$. This proves Conjecture 14.5 in Dauvergne, Ortmann, and Virág, 2022 in the affirmative.

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case where $\{\xi(z): z\in \mathbb{Z}^d\}$ are independent and identically distributed Pareto random variables with parameter $\alpha$, and assume that the solution is initially localised at the origin. We establish the full asymptotic behaviour of the total mass of the solution as time tends to infinity in the heaviest-tailed regime $\alpha\in(d,2d)$. In particular, we find qualitatively different behaviour in dimension $d=1$ and in dimensions $d\ge 2$.

We study a support-collapse version of graph pebbling on paths. A configuration is stackable if a sequence of legal pebbling moves can produce a nonzero configuration supported on a single vertex. On the path P_n, we choose a configuration uniformly from all weak compositions of total n times mu_n, where mu_n is a positive integer. We prove a two-sided fixed-offset transition for the logarithmic density. The transition is centred at sqrt(log_2 n) - (1/2) log_2 log_2 n + log_2(3e). For every fixed epsilon greater than zero, the stackability probability tends to zero when log_2 mu_n is eventually at most the centre minus epsilon, and tends to one when it is eventually at least the centre plus epsilon. No assertion is made at zero offset. The proof uses an exact recursive stackability score on trees, a one-dimensional path-message recurrence, binary-partition asymptotics for rare dyadic deficit excursions, a constant-cost regeneration argument, and an exact deep-message necessity theorem. Conditioning independent geometric occupancies on their sum returns the uniform fixed-total model. The finite deterministic necessity theorem and its exact fixed-total corollary are formalised in Lean and registered with Palomar; the full probabilistic asymptotic theorem is not part of that registration.

In this paper, we introduce a deep truncated forward-backward stochastic differential equation (FBSDE) method for high-dimensional partial differential equations (PDEs). Compared with existing deep-learning solvers for fully coupled FBSDEs, where strong coupling may lead to numerical instability, our approach exhibits improved stability. The proposed method combines gradient-truncated iterative decoupling with fictitious-play averaging to separate the forward and backward processes in a coupled framework. This preserves the coupled dynamics while reducing the unstable feedback induced by parameter-dependent forward paths during optimization. Furthermore, we incorporate a pathwise consistency term to create explicit local gradient shortcuts, thereby providing a structural mechanism that may mitigate gradient vanishing. We also derive a residual-based error estimate and establish conditional convergence of the fully discrete numerical approximations under suitable conditions, in which the pathwise consistency loss is not required. Our approach is particularly effective for convection-dominated equations, where the coupled formulation provides a stable representation of nonlinear transport without introducing singular terms into the BSDE. Numerical experiments demonstrate improved accuracy and stability in both low- and high-dimensional problems and robust performance for strongly coupled problems.

We study preservation of finite gauge-energy flows under local network repair after Bernoulli edge failures. A macroscopic demand network records terminal pairs to be routed, while a microscopic physical network contains local backup routes, bypasses and shared corridors. We prove a deterministic gauge-energy repair theorem: if the usable demand network $\mathcal{B}^{\sharp}$ carries a finite $\Phi$-energy flow $\theta$, then the repaired physical network $H$ carries a lifted finite $\Phi$-energy flow $\Theta$ with $$ \mathcal{E}^\Phi_H(\Theta) \le L\,\beta_\Phi(K)\,\mathcal{E}^\Phi_{\mathcal{B}^{\sharp}}(\theta), $$ where $L$ bounds route length, $K$ bounds routing congestion, $\mathcal{E}$ marks energy, and $\beta_\Phi$ is the gauge dilation constant. We then convert this comparison into probabilistic repair criteria: finite-dependent local repair is handled via domination by product measures, and random repair lengths via a variable-cost formulation compatible with chemical-distance estimates. As a main application, we prove a finite-dependent local bypass theorem: any macroscopic network whose supercritical percolation cluster supports a finite gauge-energy flow remains gauge-energy stable after bounded-range local reinforcement, provided the local repair probability is sufficiently high. This yields reinforced lattice and wedge-type examples and provides a potential-theoretic framework for random network repair beyond tree-like or edge-disjoint constructions.

We propose a general characterization of the hazard rate and reversed hazard rate stochastic orders for random variables that are absolutely continuous with respect to a common dominating measure. This framework is useful in giving a unified treatment of continuous, discrete, and mixed distributions without requiring ad hoc approximation techniques or restrictive integrability assumptions. Using this result, we provide direct proofs of the bivariate characterizations of both orders. Additionally, we introduce the class of $\overline{G}$-IFR and $\overline{G}$-DRHR distributions on additive groups, unifying aging properties across different domains and proving their closure under convolution. Finally, we revisit and extend the classic results of Shanthikumar and Yao (1991) on the preservation of hazard rate orders under random sums, simplifying the underlying conditions and accommodating discrete and mixed sum components.

We resolve the Rademacher singularity conjecture: an $n\times n$ matrix with independent uniform $\{-1,1\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\subset\mathbb{R}$ of cardinality $q\ge2$, we prove $\mathbb{P}(s_n(M_n)\le z/\sqrt{n}) \le Cz+a_n(S)+C'n^{1+\varepsilon}q^{-n}$ for all $n\ge1$, $z\ge0$, and $\varepsilon>0$, with $C=C(S)$ and $C'=C'(S,\varepsilon)$. Here $s_n$ denotes the least singular value, and $a_n(S)$ sums the probabilities of zero rows or columns and equal or opposite pairs of rows or columns. The proof combines inversion of randomness with Fourier averaging at scales within a polynomial factor of $q^n$.

We establish functional scaling limits and study first separation times for the interacting two elephant model studied by Aguech and Qin. In the joint diffusive regime, we give a direct martingale proof of convergence to a two-dimensional continuous Gaussian process represented by a matrix-kernel analogue of the noise-reinforced Brownian motion. We then investigate the difference process, whose diffusive scaling persists in the symmetric case even when the joint walk is critical or superdiffusive. For the first separation time of the two walks, we establish convergence in distribution to the first exit time of the limiting Gaussian process, together with convergence of all positive moments under diffusive scaling. We also obtain monotonicity results for the limiting exit time by combining explicit covariance identities with Anderson's inequality.

We fully describe the set of regular and irregular boundary points for a general hypoelliptic stochastic process $(X(t))_{t\geq0}$ absorbed outside a $C^{3,\,1}$ domain $\Omega$. The well-posedness of the associated boundary-value problem on $\Omega$ is also established in the classical sense. Let us emphasize that, throughout this work, no assumption is made regarding the inherent structure of the boundary of $\Omega$. In particular, characteristic and non-characteristic points may coexist on $\partial\Omega$, thus extending previous results obtained in the literature. We show that, unlike previous settings, the solution to this boundary-value problem may exhibit discontinuity at the boundary $\partial\Omega$. In certain cases involving a $C^\infty$ domain $\Omega$ and $C^\infty$ coefficients, the discontinuity set is even of positive Lebesgue surface measure on $\partial\Omega$. Finally, we highlight a cone condition under which the continuity is preserved at the boundary.