Papers

We analyze the level lines of the Gaussian Free Field with logarithmic singularities in polygonal domains. We focus on the (none-zero) singular case and compare it to the singular-free setting. The law of the level lines for the field in singular case is absolutely continuous with respect to the law for the singular-free case, with a Radon-Nikodym derivative related to the reduced Green's energy which is introduced by F. Viklund and D. W. Nyström~[VN24]. These level lines admit a Brownian motion representation. In the singular-free case, such representation involves Poisson point processes of Brownian loops and excursions; while for the singular case, it additionally includes Poisson point processes of Brownian bubbles and bridges. The special cases of one or two singularities yield moment-generating functions of the conformal radius, which can be interpreted as one-point and two-point correlation functions of vertex operators in conformal field theory at the free-boson point. While the Radon-Nikodym derivative is bounded away from the singularities, the behavior of the level lines near the singularities is markedly different from the singular-free case, with different asymptotic probabilities for all level lines to remain close to the singularity.

Let $1\le k\le n$ and let $A$ be the adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices. Suppose that $\lambda n\le d \le (1-\lambda)n$ for a fixed $0<\lambda\le 1/2$. We show that there exists $c_{\lambda}>0$ depending only on $\lambda$ such that $$ \mathbb{P}[\operatorname{rank}(A)\le n-k]\le 2e^{-c_{\lambda} kn}. $$ This gives a large corank extension of the exponential singularity bound of Jain, Sah, and Sawhney.

We study continuous-time supercritical branching random walks in space-inhomogeneous random environment on the Hamming cube $\{0,1,\dots,b-1\}^d$, where the reproduction laws at each site are i.i.d. sampled. This serves as an idealized model for RNA sequence evolution with mutation–selection balance. The reproduction and mutation events are decoupled, and we assume that the reproduction law has essential supremum strictly below the deterministic mutation rate. Our main results provide tight asymptotics of the first-passage times for the model, uniformly in the origin–target distance $1\le m\le d$ as $d\to\infty$, conditional upon survival. We prove both quenched tightness in environment probability and annealed tightness around a deterministic center. Moreover, we identify the leading order of the deterministic center and develop an expansion in the sparse-distance regime $m=o(d)$. Our proof technique derives quantitative approximations of the model using an independent-visit variant, where revisits still resample the environment. As an application, we show that on a macroscopic scale, increasing the reproduction law in convex order decreases the first-passage times, and increasing the mutation rate increases the first-passage times for the sparse regime.

We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a $p_n \times n$ data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime $p_n / n \to \phi \in (0, \infty)$, the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is $k_* (\phi) = \lfloor \sqrt{\phi} \rfloor + 2$, yielding the critical tail exponent $\alpha_* (\phi) = 2 + 2 / k_* (\phi)$. This exponent decreases in steps as $\phi$ increases, creating a staircase boundary between convergence to the upper Marčenko–Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.

We design the first polynomial-time algorithms for approximately counting and almost uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the study of quantum many-body systems, and beyond. Our approach has two key ingredients. $\bullet$ We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "$\textit{soft intersection}$" satisfies a Poincare inequality uniformly over all external fields. $\bullet$ We solve a dual maximum entropy convex program to compute external fields under which the hard constraint is satisfied with high probability under the soft intersection measure. We bound this success probability directly using the uniform Poincare inequality and smallness of the gradient norm. $\textbf{AI Disclosure}$ GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper. A more complete discussion is included in the acknowledgments.

We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields $(Y,Z)$, where $Y$ is a functional on the space of probability measures and $Z$ is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field $p$. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify $Z$ with both the characteristic field $p$ and the intrinsic gradient of $Y$. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component $Y$ solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, $Y$ is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.

On the set $\mathbb{YF}_n$ of words in $1$ and $2$ with digit sum $n$, we consider the probability distribution $\mu_P$, corresponding to the Plancherel measure on paths in the Young–Fibonacci graph. We study the asymptotics as $n$ grows of the entropy of the distribution $\mu_P$ and of the minimum number of words on which $\mu_P$ is almost entirely concentrated.

We study upper large deviations of the free energy of mixed $p$-spin Ising spin glasses through the generalized TAP free energy $F_{\mathrm{TAP}}$ of Chen, Panchenko and Subag. For every mixture with radius of convergence greater than one, $\max F_{\mathrm{TAP}}$ and $\log Z_N$ have the same large deviations at speed $N$; the proof combines their band theorem with Ramsey's theorem. For convex mixtures with $\sum_p 2^p\beta_p^2<\infty$, upper deviations at level $f$ are carried by generalized TAP critical points, which exist at the free-energy rate without Boursier's strict Plefka condition. Outside an event of smaller exponential order, near-maximizers at level $f$ lie near the contact set of a constrained Parisi obstacle, follow the Auffinger-Chen field law, and have Hessian bulk near a reflected Pastur law whose edge is nonpositive and vanishes exactly at contacts where the obstacle is flat to second order. Where every contact is of this kind, a fixed stability margin costs rate.

Let $Z_N$ be the partition function of the Sherrington-Kirkpatrick model at inverse temperature $\beta>1$. Replica theory (Kondor near the critical temperature, Parisi and Rizzo at all temperatures) predicts that the probability that $N^{-1}\log Z_N$ exceeds its typical value by $\delta$ decays like $\exp(-Na\delta^{6/5})$ with an explicit $a$; Aronow and Lopatto recently proved bounds of this order. The exponent is the Legendre transform of the Parisi problem constrained to carry an atom $\theta$ at zero. We show that the constrained minimum exceeds the equilibrium free energy by $c_\beta\theta^5(1+o(1))$, where $c_\beta=\frac{9}{640}\beta^3\rho_\beta(0)^{-3}$ and $\rho_\beta(0)>0$ is the density at zero of the Parisi measure. With the fractional-moment formula this gives the exponent $\frac56(6c_\beta)^{-1/5}\delta^{6/5}(1+o(1))$ as $N\to\infty$ and then $\delta\to0$. The constrained minimizer has a marginal first positive contact, to leading order at $3\theta/(2\rho_\beta(0))$ with an atom $\theta/2$. This plateau calculus applies to every mixture with $\xi'''(0)=0$ whose Parisi measure accumulates at zero; a small field $h$ opens a gap of order $|h|^{2/3}$, and an atom of the Parisi measure at zero, present in every pure $p$-spin model with $p\ge3$, makes the rate leave zero linearly. Near criticality the constant tends to Kondor's value $9/5120$ and the marginal regime ends at a de Almeida-Thouless threshold. For the Sherrington-Kirkpatrick model at every $\beta>1$, a finite-step curvature inequality from an unrefereed preprint shows that every constrained minimizer either has an interval of positive contacts, all marginal, or has exactly two atoms, and that the marginal regime ends at a unique de Almeida-Thouless threshold $\theta_c(\beta)$. The laws at every $\beta>1$ use that the Parisi measure accumulates at zero, a theorem of Lopatto also proved in earlier work of the author.

We provide a complete study of the optimal control problem associated to the so-called Dyson equation, which describes the mean field evolution of the spectrum of large random matrices. We treat both the case of regular and singular terminal data. We construct admissible controls and prove the existence of optimal ones. We then proceed to give a thorough mathematical analysis of the associated Hamilton-Jacobi-Bellman equation by means of the theory of viscosity solutions in the space of measures. In particular, we characterize the value function of the optimal control problem as the unique viscosity solution to this infinite dimensional HJB equation.

We study empirical processes of observable extreme generators in stationary Poisson-Laguerre tessellations, with each generator weighted either by one or by the volume of its cell. Under moment assumptions that allow heavy-tailed marks, we prove functional central limit theorems as the observation window grows, on compact intervals where the mark law has a bounded density. We also obtain covariance asymptotics with a surface-order error and quantitative univariate and multivariate Gaussian approximation.

In the continuity of [1, 2, 3], we study a class of constrained mean-field Langevin dynamics formulated as a Wasserstein gradient flow with an energy constraint. We show that the constrained evolution therefore defines a well-posed dissipative dynamics in the Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$. We prove that the flow preserves probability mass and the energy constraint, dissipates the free energy monotonically, and converges exponentially to a unique constrained equilibrium under geodesic convexity assumptions. Using synchronous coupling, under Lipschitz assumptions, we demonstrate the quantitative $L^2$-propagation of chaos with a Fournier-Guillin bound, and under conditions of sufficient dissipativity, we derive an explicit time-uniform bound. We also demonstrate exponential controls in the $\mathcal{W}_2$ Wasserstein metric and the contraction of the invariant measure map. Using log-Sobolev inequalities, we show that the mean-field entropies decay exponentially along the constrained flow. The results are applied to several concrete and original examples. Keywords and phrases: Wasserstein gradient flows $\bullet$ mean-field Langevin dynamics $\bullet$ constrained nonlinear Fokker-Planck equations $\bullet$ Lagrange multiplier $\bullet$ JKO scheme $\bullet$ propagation of chaos $\bullet$ synchronous coupling $\bullet$ exponential ergodicity $\bullet$ functional inequality $\bullet$ log-Sobolev inequality $\bullet$ transport inequality $\bullet$ Talagrand T 2 -inequality

We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension $d\geq2$. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free $L^\infty_t{\rm BMO}^{-1}_x$ drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in $L^\infty_tL^d_x$. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent $L^{d,\infty}$ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent $L^d$ drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.

This paper introduces a robust strategy to establish large deviation principles for mean field interacting particle systems in T d with vanishing noise. Viewed as a Freidlin-Wentzell type problem, our approach relies on recent techniques for partial differential equations (in short PDE) on the space of probability measures [5]. We first outline the proof strategy for the PDE approach to large deviations in a self-contained manner. As the number of particles goes to infinity, we then show that the transfer function of the system converges to the unique viscosity solution of a limiting Hamilton-Jacobi equation on the space of probability measures. Finally, we explicitly derive the rate function by formulating the solution of this limiting PDE as the value function of an optimal control problem.

We study strong completeness and the existence of a set attractor for the stochastic flows induced by a class of two-dimensional stochastic differential equations resembling a planar Ornstein-Uhlenbeck process with an additional radius-dependent rotational drift term. Our main results give both sufficient and necessary conditions for the stochastic flows to be strongly complete and for the existence of set attractors. In particular, we demonstrate that if the derivative of the angular velocity $\rho(r)$ with respect to the radius $r$ satisfies $|\rho'(r)| \geq K_2\, r^3$, for large $r$ and some universal constant $K_2$, the diameter of a compact set can grow exponentially fast with positive probability. Furthermore, we show that for $|\rho'(r)|\geq r^{3+\varepsilon}$, $\varepsilon>0$, with probability one, exceptional initial conditions diverge to infinity in finite time, ruling out the existence of a global stochastic flow.

We investigate two learning procedures and their relationship to the formation of equilibrium in potential mean field games (MFGs). The first one is a version of the fictitious play in which players observe directly the actions (controls) of other players and learn the distribution of the MFG through an update rules on controls. This leads to a new iterative procedure which converges for a wide class of potential MFGs of controls, even in the presence of common noise. On the other hand we also consider the learning procedure of a principal agent trying to solve a mean field control problem with only local gradient information. We prove that under very general assumption, the gradient descent of such agent converges to a solution of the associated potential MFG.

The binary perceptron model studies the set of solutions obtained by intersecting random Gaussian half-spaces with the discrete hypercube {-1,+1}^n. Pioneered in the work by Krauth and Mézard in the 1980s [KM89], one of the most striking physics predictions for this model is strong freezing, which says that typical solutions are isolated. Moreover, the conjecture was sharpened later to assert that typical solutions are at Hamming distance of order $n$ from every other solution [HWK13],[HK14]. This phenomenon has attracted a lot of attention over the years, partly due to its intricate connections to statistical physics and computational complexity. We prove that the binary perceptron model exhibits strong freezing with high probability at every fixed positive constraint density and every fixed margin. Our proof is based on the planted model and a restricted partition function that counts solutions having nonnegative overlap with the planted solution. Unlike the full partition function, this restricted one is monotone in the planted direction, so the FKG inequality decouples it from isolation, while the restriction costs only a factor of n+1. Together, these allow us to transfer the isolation of the planted solution to typical solutions, which completes the proof of strong freezing.

The goal of the present paper is to investigate the exact decay rate of the tail probability $\mathbb{P}(|X_1-x_0|>R)$ for large $R$, where $X_t$ is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition $x_0$. In the first place, under the assumption of uniform ellipticity, we establish a general $(2H+1)$-Weibull lower tail estimate in Young's regime of $H\in(1/2,1)$ and a general Gaussian lower tail estimate in the rough regime of $H\in(1/4,1/2)$. These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with $C_b^\infty$-coefficients could fail to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any $2\times 2$ periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general $(2H+1)$-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.

We consider Walsh's process on a star graph perturbed by the fact that its paths need to additionally pass through semi-permeable soft membranes, each located on a different edge at a small distance from the center, and each having its own permeability coefficient. We describe the limit processes as the distances converge to $0$, in terms of the characteristics of the approximating ones. The main finding is that the limit process is still of Walsh type, but its parameters need to be appropriately modified. The paper is a close companion to A. Bobrowski and A. Pilipenko “On Portenko's approximation of skew Brownian motion” (arXiv:2601.20440), and to A. Pilipenko and G. Vilmart, “Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits.”

We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, multiplied by $\sqrt{\log n}$, converges stably relative to the full Gaussian field without additional centering. Conditionally on that field, the limit consists of Fourier coefficients of one symmetric complex stable random measure of index one with control proportional to $M$. For every $s>1$, the normalized modulated measures also converge stably in $H^{-s}(\mathbb{T})$ to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most $(\log\log n)^{1/64}$ for every fixed admissible covariance. The scalar limit is an isotropic Cauchy mixture; the joint limiting law retains the spatial distribution of $M$, not only its total mass. We also prove an almost-sure integral test along the entire integer frequency sequence. For a class of regular deterministic gauges $H$, the limsup of $\sqrt{\log|n|}\,|C_n|/H(\log\log|n|)$ is zero or infinity according as $\int^\infty dx/H(x)$ converges or diverges. The proof combines index-one compensation with relative excursion estimates and transfers these conclusions from the canonical model by smooth covariance comparison.

Let $A$ be the (non-symmetric) adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices, and let $\Xi$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min(d,n-d)\ge\lambda n$ for some fixed $\lambda\in(0,1/2]$. We show that there exists $c > 0$, depending only on $\lambda$, such that \[ \mathbf P_{A,\Xi}\{\operatorname{rank}(A\circ\Xi)\le n-k\}\le e^{-c nk},\qquad 1\le k\le n. \] As an ingredient in the proof of the rank bound, we use the case $\kappa=0$ of the following quantitative smallest singular value estimate: \[ \mathbf P_{A,\Xi}\{s_n(A\circ\Xi)\le\kappa\} \le C\kappa\sqrt n+e^{-c' n}, \qquad \kappa\ge0, \] where $C,c'>0$ depend only on $\lambda$.

We establish large and moderate deviation principles for a broad class of nonlinear stochastic partial differential equations driven by multiplicative Poisson random measures in the small-noise regime. Our abstract framework is formulated for locally monotone evolution equations over a Hilbert triple and encompasses several important models arising in fluid dynamics and turbulence theory, including the two-dimensional Navier–Stokes equations, magnetohydrodynamics (MHD), the GOY shell model of turbulence, the two-dimensional tidal equations, and Navier–Stokes equations with nonlinear viscosities. A key analytical ingredient is the well-posedness of the associated controlled equations, established using the method of local monotonicity. The large and moderate deviation principles are then proved via the weak convergence approach.

We give two-sided moment estimates for decoupled chaoses of any order generated by independent symmetric random variables with regular moment growth: $\|X\|_{2p}\le K\|X\|_p$ for every $p\ge1$. The estimates involve only the coefficient tensor and the one-dimensional tails. They use the partition norms of Adamczak and Latała, with constants depending only on the order and on $K$. The proof starts with log-concave tails. Gaussian smoothing gives the estimate for the expected norm needed for induction, and a product comparison reduces the remaining distributions to this case. The extra tensor indices introduced by the comparison are removed using Rademacher moments. A comparison with products of log-concave-tail variables then extends the formula to regular moment growth. We also give tail bounds and explicit Rademacher and Weibull formulas for $r\ge1$, including fourth-order examples. This article was developed with the assistance of GPT Astra-6.

We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on $\mathscr E=\{u\in H^1\cap L^8: \partial_zu\in L^4\}$ under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the $H^1$ topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable–compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy–cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.

Approximate solutions are found for Feller-like neutral multi-type branching diffusions $\big{(}\mathbf{X}(t)\big{)}_{t \in \mathbb{R}_{\ge 0}}$ in the limit of small mutation rates. The method employed involves solving approximations to the Laplace transformed forward Kolmogorov equation by integrating along characteristics. To leading order in the scale $\theta$ of the overall mutation rate the super-critical diffusion is found to collapse onto a line density aligned with the stationary left eigenvector of the rate matrix following a rapid change of behaviour at a critical time $t_{\rm c}$, which has a weak logarithmic dependence on $\theta$. First order in $\theta$ approximations to the density and moments of $\mathbf{X}(t)$ are also determined. The first-order approximation corresponds to allowing at most one mutation in the coalescent tree of a sample and is valid for $t < t_{\rm c}$ in the supercritical case, and more generally for the critical and sub-critical cases.

We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.

We study state-constrained stochastic control problems with a scalar surplus $R \geq 0$. Controls for which the drift and volatility of $R$ vanish at zero define a lower-dimensional Hamilton-Jacobi-Bellman equation. Starting from control-wise state-constraint viscosity inequalities, we prove that the upper and lower limits of the interior value are a subsolution and a supersolution of this boundary equation. Terminal compatibility and comparison identify their common limit. We require one-sided bounds on positive surplus drift and generator growth, together with domination by boundary generators. For compact or coercive controls, these follow from a local lower bound and continuity of the boundary control set. For a smooth stochastic target value $w$, the transformation $R = Y - w(t,X)$ flattens the viable epigraph, and, under the stated hypotheses, the boundary control value supplies the Dirichlet datum. Applications include set-valued boundary controls, unbounded drift with quadratic costs, and redundant hedging instruments with a nonconstant target boundary.

We study the offline gap between deterministic calibration distance C and its fractional relaxation L for binary unit-weight sequences under total absolute-change cost. We sharpen the offline comparison C <= L + O(sqrt(T)) (Qiao and Zheng, 2024, Theorem 2) to the sharp worst-case order Theta(T^(1/3)). If Delta_T is the supremum of C - L over length-T inputs, then T^(1/3)/1000 <= Delta_T <= 41T^(1/3) for T >= 216. The upper bound holds for every input, while each T >= 216 has a rational lower-bound input. For every input with m distinct forecasts, C <= L + m, and the unrestricted-sample worst-case sparse order is Theta(m). For rational forecasts and accuracy, with binary-encoded multiplicities of separately assignable unit identities, a grid-free polynomial-bit-time procedure returns B <= L <= U, U - B < eta, and an exactly calibrated compact repair of cost at most U + m <= L + m + eta.

Feller random measures generalize the Feller diffusion (the CIR process) by giving it memory. They arise as the scaling limits of nearly unstable Hawkes processes, and include the rough CIR process and its hyper-rough and discontinuous relatives. We show that every Feller random measure is the occupation measure of a Dawson–Watanabe superprocess whose spatial motion is a killed Lévy subordinator, integrated over the branching time. This is a continuum analogue of the Hawkes–Oakes cluster representation. It yields existence, stability in the parameters, and stochastic equations driven by an explicit noise. The structure of this noise depends on whether the kernel has an atom at the origin. Without an atom, the noise is a Brownian motion time-changed by the distribution function of the measure. This leads to a martingale characterization and to sharp results on densities and their Hölder regularity. With an atom, the noise is a compensated inverse-Gaussian process time-changed by the compensator of that distribution function, and the measure is purely atomic.

We give non-asymptotic operator-norm tail bounds for random multi-matrix models with quadratic plus globally Lipschitz interactions. More precisely, let $\mu_f$ be the probability measure on $(\mathbb{M}_n)_{\operatorname{sa}}^m$ with density proportional to $\exp(-n^2(\lVert x \rVert_2^2/2+f(x)))$. If $f$ is $L$-Lipschitz with respect to $\lVert \cdot \rVert_1$, we construct a coupling of $X\sim\mu_f$ and an $m$-tuple $Z$ of independent GUE matrices such that $\lVert X-Z \rVert_\infty\leq L$ almost surely, where $\lVert \cdot \rVert_\infty$ is the maximum of the operator norms of the matrix tuple. This is an operator-norm analog of the $L^\infty$-Wasserstein bound of Khudiakova, Maas, and Pedrotti (2025). We give a new proof based on the variational formula of Boué and Dupuis (1998), the associated Hamilton–Jacobi–Bellman equation, and the duality between entropy and pressure.