Papers

We study the sign structure of Ursell functions in the Blume–Capel model. For $\Delta \le \log 2$, we give a short proof of the alternating sign property using an Ising representation. We then construct counterexamples for every $\Delta > \log 2$, showing that the condition $\Delta \le \log 2$ is sharp.

We introduce a fully stochastic, non-Markovian SIRS-type epidemic model that incorporates varying infectivity, waning immunity and a pulse vaccination strategy that may not confer permanent immunity. The model is constructed at the individual level, where each person is characterized by random infectivity and susceptibility functions, and vaccination campaigns occur at the jump times of a Poisson random measure with arbitrary intensity. We rigorously derive the epidemic dynamics as the large-population limit of an interacting stochastic particle system, leading to a system of nonlinear Volterra-type integral equations governing the average susceptibility and total force of infection. We establish a functional law of large numbers(FLLN) for the empirical processes and provide explicit expressions for the limiting compartmental proportions. The long-term behavior of the system is analyzed: we prove that the infection-free solution is globally asymptotically stable when the basic reproduction number falls below a critical threshold, and that the disease persists when this threshold is exceeded. The threshold is given by the harmonic mean of the maximal susceptibility across individuals and generalizes previous results by incorporating vaccination and memory effects. Our framework provides a probabilistically grounded extension of classical deterministic pulse vaccination models and offers new insights into the control of epidemics through scheduled immunization policies.

Chronic wasting disease (CWD) is a fatal prion disease affecting deer, elk, moose, reindeer, muntjac, and other cervids. Because free-ranging cervid populations face environmental variability and randomness, deterministic models may miss important dynamics like stochastic fade-out. We develop a stochastic Susceptible-Infectious-Environmental model using differential equations with reflection to ensure the susceptible class remains non-negative. We examine how environmental variability influences cervid populations as CWD pressure and control measures increase. For the deterministic model, we derive the basic reproduction number as the sum of direct and environmental contributions, showing the endemic phase arises at R0=1. For the stochastic system, we establish local well-posedness, positivity, and the disease-free law. The top Lyapunov exponent for invasion remains unaffected by reflection. We evaluate CWD mitigation using a deep reinforcement learning agent trained with Proximal Policy Optimization in a hybrid action space, comparing hunting, decontamination, and combined strategies. In the deterministic case, hunting alone can control the disease but reduces the population by about 58%, while decontamination requires sustained effort. The combined policy more than doubles the cervid population and nearly eliminates infection and contamination. In the stochastic case, the policy contains the disease in about 80% of runs, with 10% experiencing large outbreaks; effectiveness decreases as noise increases. Across all scenarios, the agent consistently emphasizes environmental decontamination, the key control method.

We study an asynchronous consensus dynamics on $N$ agents: at each step a uniformly chosen agent replaces its state by $f(Y_1,\dots,Y_r)$, where $f$ is a fixed monotone aggregation rule and $Y_1,\dots,Y_r$ are the states of $r$ agents sampled uniformly with replacement. Let $T$ be the first time at which all agents agree. Let $f:\{0,1\}^r\to\{0,1\}$ be monotone and non-constant. The expected consensus time is governed by the two endpoint degrees $D_0(f)=\#\{i:f(e_i)=1\}$ and $D_1(f)=\#\{i:f(\mathbf{1}-e_i)=0\}$, where $e_i$ is the $i$th standard basis vector and $\mathbf{1}$ the all-ones vector. If $D_0(f)\ne 1$ and $D_1(f)\ne 1$, then $E[T]=\Theta\bigl(N(1+\log d_f)\bigr)$, where $d_f$ counts the agents that must change state before the nearest attracting consensus is reached, so that $E[T]=O(N\log N)$ uniformly over initial states. If $f$ is a dictator, $E[T]$ is given by a voter-model formula and equals $\Theta\bigl(N^2 Ent(p_0)\bigr)$ with absolute constants, uniformly over the initial state, where $Ent$ is the binary entropy and $p_0$ the initial fraction of agents in state $1$. Otherwise exactly one of $D_0(f),D_1(f)$ equals $1$ and the other equals $0$. The worst-case expected consensus time is then $\Theta(N^{2-1/m})$, where $m\ge 2$ is the least number of coordinates that force the value $1$ in the residual rule of $f$, defined in the paper (of its dual, when $D_1(f)=1$).

In this paper, we provide an alternative proof of a result due to Gross (1960s), extending it to the Fréchet space setting. Our argument adapts the proof technique developed by Bogachev in the Banach case, as presented in the monograph Gaussian Measures, to the more general framework of separable Fréchet spaces, simplifying several steps along the way.

This paper establishes the algebraic and categorical foundations of spectral duality structures (SDS). We show that the class of all SDS admits the structure of a graded symmetric monoidal category, where the degree $K$ is the fundamental invariant indexing the strata. We define operations of Cartesian product and disjoint union, morphisms preserving involution and weights, and prove that $C^\ast=1/(1+\sqrt K)$ is a functor constant on each stratum. We classify objects up to isomorphism by the combinatorial type $(k,f)$, the degree $K$, and a multiset of inversion classes $[r]=\{r,r^{-1}\}$. In each degree, skeletons are exactly the finite $C_2$-sets over the additive group of the reals with $C_2$ acting by $t\mapsto -t$, which explains the monoidal structure and identifies the classification with the decomposition into orbits. Furthermore, we prove that the response rank is the rank of an explicit matrix built from the amplitudes and the weights, and hence exactly the number of non-trivial pairs $k$ under the full-rank condition, and we connect the categorical structure with the Fisher–Rao geometry of Paper V. The SDS, originally discovered in stochastic resetting problems, is thus an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.

Let $X\sim Bin(N,p)$ and \(Y=|X-Np|\). We derive an exact reduction for every odd absolute moment of \(Y\), expressing it through finitely many central masses and central tail probabilities. The tail coefficients satisfy a sum rule and are divisible by \(q-p\), so that in the symmetric case the odd ladder collapses to masses alone. We then obtain the complete central-tail expansion at a bounded lattice displacement, with Bernoulli-polynomial coefficients in integer powers of the large parameter. As an application, the first two tail coefficients yield a second-order expansion for the median of the beta distribution, whose formal one-parameter degeneration reproduces the first two terms of Choi's expansion for the gamma median.

We determine the Boltzmann–Grad limit of the Ehrenfest “wind-tree” model with a periodic configuration of polyhedral scatterers. The central result is a loss-of-memory effect in dimension $d\geq 3$ for a generic choice of polyhedron. This yields a Markovian limiting transport process on an extended state space and a generalised linear Boltzmann equation for the time evolution of the particle density.

We derive nonasymptotic upper bounds on the probability that a generalized current of a geometrically ergodic diffusion observed for any amount of time, or its sample mean over any arbitrary sample size, deviates from the stationary mean by more than any given amount. The concentration-of-measure behavior of generalized currents is universally governed by the relaxation time of the underlying dynamics, the locally observed dissipation rate, and the intrinsic local fluctuations of the observable. We uncover stark qualitative and quantitative differences in fluctuations in and out of thermodynamic equilibrium. We further obtain refined inverse thermodynamic uncertainty relations, bounding the variance of generalized currents from above. We construct nonasymptotic confidence intervals for controlling uncertainty in thermodynamic inference from small data, i.e., from short trajectories and small samples, and provide the first quantitative answer to when a trajectory is sufficiently long and a sample is sufficiently large. As an illustration, we apply our results to currents observed on a two-dimensional Ornstein-Uhlenbeck process in and out of equilibrium and show how they can be used to rigorously detect broken detailed balance.

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $\kappa=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_\kappa$ as $\kappa \downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_\kappa$ graph metric converges as $\kappa \downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

We continue our study of the conformal loop ensemble (CLE) with parameter $\kappa=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.

We consider the conformal loop ensemble (CLE) with the parameter $\kappa=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_\kappa$ loops as $\kappa \downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.

Probability plays two distinct roles in modern data-analytic practice: (i) as an internally coherent, filtration-relative language for sequential forecasting and, together with a stated loss or utility, decision making; and (ii) as a foundation for empirical claims such as stabilization, calibration, and repeated-sampling validity. The first role is relative to an assessment law $\mathbb Q$, whereas the second is evaluated under a governing law $\mathbb P$ and a declared repetition regime. In classical settings the roles are often aligned by conditional correctness under $\mathbb P$ together with i.i.d.\ or other stable-law assumptions under a fixed design, or by exchangeability. In adaptive and AI-mediated settings they can diverge because forecasts operate within feedback loops while reliability is evaluated under a potentially different law and regime. We synthesize relevant results from prequential forecasting, calibration, martingale and game-theoretic validity, adaptive data analysis, conformal prediction, and long-context AI reliability into a formal account of role separation. The resulting Role Separation Principle distinguishes internal coherence from conditional-mean correctness under the evaluation law and from target-specific reference and stability conditions. Two separation results show that a proper coarsening of the limiting-frequency distribution need not identify the governing law, and that even conditionally correct forecasts under that law need not accompany stabilization. We then develop a unified stabilization framework spanning filtration-based, shift-based, policy–environment, and design-based regimes and operationalize it through a regime-conditional audit checklist. Applied illustrations clarify calibration claims, synthetic-data validity, and structural mismatches relevant to some high-confidence unsupported AI outputs.

We give deterministic FPTASes for two dense counting problems on which the known deterministic algorithms, based on zero-free interpolation, run in quasipolynomial time. For fixed $0<\gamma<1/2$ and $0<\theta\leq1$, the first approximates $\mathrm{haf}(A)$ for a symmetric matrix $A$ when its support graph $G$ has minimum degree at least $(1/2+\gamma)n$ and its nonzero entries lie in $[\theta,1]$. It also approximates permanents under the analogous bipartite condition, including full-support matrices in $[\theta,1]$. For fixed $\beta>0$ and $0<\kappa\leq1$, the second approximates the zero-field Ising partition function $Z(J)$ for zero-diagonal real symmetric matrices $J$ satisfying $\max_{i,j}|J_{ij}|\leq\beta/n$ and $\lambda_{\max}(J)\leq1-\kappa$. No separate lower-eigenvalue condition is imposed. We further prove $\log\mathrm{haf}(A)=h_A(G)-n/2+O_{\gamma,\theta}(1)$ and $Z(J)=2^n\det(I-J)^{-1/2}(1+O_{\beta,\kappa}(1/n))$. Here $h_A(G)$ is the maximum weighted fractional-matching entropy. For unweighted graphs, the first formula improves the Cuckler–Kahn error from $o(n)$ to $O_\gamma(1)$ on the fixed-margin class and extends it to weights in $[\theta,1]$. Both algorithms use a common Gaussian truncation principle. Each problem becomes an integral of a product of a fixed entire function over Gaussian coordinates, with possibly indefinite moment matrix entries of order $1/n$. Cancelling the linear term and exactly resumming the quadratic term leaves a coordinate remainder vanishing to order at least three. Complex dilation handles small supports. For large supports, we bound the recombined tail by a large-deviation rate that beats the entropy of the subsets. The truncation error is at most $(CR/n)^{R/2}+e^{-cn}$. This faster-than-geometric decay permits $R\log(en/R)=O(\log n+\log(1/\epsilon))$ and hence polynomial enumeration.

We investigate limits of semigroups of stochastic matrices defined by their invariant distribution. Given probability vectors $\gamma(\beta)$ depending on a parameter $\beta$, we introduce a notion of convergence as $\beta\to\infty$ for the corresponding semigroups of $\gamma(\beta)$-preserving stochastic matrices and investigate the structure of the resulting limit. In general, the limiting semigroup differs from the semigroup preserving the limiting distribution, showing that these two operations do not commute. We develop a general framework for such limiting semigroups and study in detail the case in which the invariant distributions are Gibbs vectors at the inverse temperature $\beta$. We show that the limiting semigroup consists of block-upper-triangular stochastic matrices subject to additional substochasticity constraints. We characterise and enumerate their extremal elements and determine the preorder on probability vectors induced by the action of the semigroup. The resulting notion of Block-Triangular majorisation interpolates between ordinary majorisation and upper triangular (aka unordered) majorisation. We show that it is completely characterised by a finite family of monotones and analyse the corresponding behaviour of Rényi $\alpha$-entropies as $\beta\to\infty$.

We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let $d\ge1$, $L\ge0$, and let $B:\mathbb{R}^d\to\mathbb{R}$ be globally $L$-Lipschitz. We prove that the Brenier map from the standard Gaussian measure $\gamma_d$ to the probability measure proportional to $\mathrm{e}^{-B}\gamma_d$ has a globally Lipschitz representative whose bound depends only on $L$. The Brenier potential also belongs to $C^{1,1}(\mathbb{R}^d)$ and satisfies dimension-free two-sided Hessian bounds. Let $\mathfrak{C}(0,L)$ denote the upper-bound constant, then \[ \log\mathfrak{C}(0,L)=4L^2+\log L+\mathcal{O}(1) \qquad(L\to\infty), \] and a one-dimensional example with Brenier map $T_L$ satisfies $\log\operatorname{Lip}(T_L)\ge L^2/2$. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let $V,W:\mathbb{R}^d\to\mathbb{R}$ be potentials, and let $Q,P$ be positive-definite matrices. Using the distributional curvature bounds $D^2V\preceq Q$ and $D^2W\succeq P$, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.

We study heat content and spectral properties of subordinated sub-Laplacians on arbitrary Carnot groups. We consider restrictions of such operators to bounded open sets with zero Dirichlet boundary condition and study their spectral properties. In particular, for a large class of subordinators we give explicit eigenvalue estimates in terms of the subordinator and the eigenvalues of the sub-Laplacian. We also provide large-time asymptotics of the heat content for the subordinated sub-Laplacian in terms of its spectral gap. For fractional sub-Laplacians we prove short-time asymptotics for the corresponding heat content and relative heat content. Our approach combines semigroup methods, probabilistic techniques, geometric measure theory, and heat kernel estimates.

We construct and characterize the one-parameter family of real-valued processes that allow to recover the law of a Brownian loop-soup of any intensity on the real line conditionally on its occupation time field. These processes generalize the Brownian burglar constructed by Warren and Yor (that corresponds to our process in the limiting case where the intensity vanishes and there is just one Brownian motion). These processes are closely related to the Bass-Burdzy flow and recent work of Aïdékon, Hu and Shi on the stochastic Jacobi flow. Our approach uses a formalism, building on the notion of driver processes, where one considers the evolving object to be the line that we “stretch", instead of the burglar itself and its remaining occupation time. This leads to a characterization of these (generalized) Brownian burglars by simple natural axioms. It also allows to treat “negative intensities” and provides fairly direct derivations of several properties of the burglars, and in particular a “target-independence/locality" property at the special intensity where the local time is the square of a Gaussian Free Field.

We identify non-Gaussian corrections to the central limit theorem for the global and local laws, i.e. for Stieltjes transform of the empirical eigenvalue distribution of a large real symmetric or complex Hermitian Wigner matrix. We find that in the real case the rate of convergence in this CLT is substantially slower than in the complex case. For the proofs, we develop new static and dynamic hierarchies of extended cumulants up to any order that can be analyzed by a refinement of the zigzag strategy.

We study the asynchronous $\ell^p$ relaxation introduced by Amir, Nazarov, and Peres: at each step, a uniformly chosen vertex minimizes its incident $\ell^p$ energy. For the profile $f_t$ after $t$ updates, let $$ \mathsf T_p(G,1/2):= \sup_{\lVert f_0\rVert_\infty\le1} \mathbb{E}\bigl[\min\{t\ge0:\operatorname{osc}(f_t)\le1/2\}\bigr]. $$ For $1<p<\infty$, we obtain graph-dependent estimates for boxes, trees, and conductance expanders. On the nearest-neighbor box $[L]^d$, for $d,L\ge2$ and $n=L^d$, the answer is, up to logarithmic factors, $nd^{1/(p-1)}L^{p/(p-1)}$ for $1<p<2$ and $ndL^2$ for $p\ge2$. On bounded-degree trees, an explicit rerooting-invariant parameter $T_G$ determines the answer up to logarithmic factors; for arbitrary trees it gives upper and lower bounds that differ additionally by the maximum degree. If the volume conductance $h(G)\ge h_0>0$, then $\mathsf T_p(G,1/2)=\Theta_{p,h_0}(n\log n)$ without a degree assumption. At $p=\infty$, every connected graph satisfies $\mathsf T_\infty(G,1/2)\ge c nD^2/\Delta$, where $D$ and $\Delta$ are its diameter and maximum degree.

In this article, we investigate the two-species zero-range process, a multi-species generalization of the classical zero-range process. First, we analyze its condensation regime, which directly parallels its single-species counterpart. As our main result, we establish the dynamical metastable behavior of the location of the single condensate, showing that its motion is governed by a simple Markov chain on the accelerated time scale $N^{1+\alpha}$, where $N$ denotes the total number of particles in the system and the parameter $\alpha>1$ governs the attractivity of the system. Another novelty of this work lies in the proof technique, which integrates the recently developed resolvent approach with the $H^{1}$-approximation method to rigorously characterize metastability.

Let $U$ and $H$ be two separable Hilbert spaces and $T>0$. We consider a stochastic differential equation which evolves in the Hilbert space $H$ of the form \begin{align} \label{SDEa} dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $G:U\to H$ is a linear bounded operator and $B:[0,T]\times H\to H_\beta$ is a locally $\theta$-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable $\theta\in(0,1)$. Here, $H_\beta$ is a Hilbert space which contains $H$ with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space $H$ into itself. The coefficient $\beta\geq0$ measures the order of such a singularity and the case $\beta=0$ corresponds to a drift term with values in $H$. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. In particular, the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler–Bernoulli beam equation up to dimension $3$, even in the hyperbolic case.

We consider the space of $L^p$-bounded martingales on the Cantor space for $p\in[1,\infty]$. Equipping the Cantor space with two standard metrics, we show that under both, a typical martingale diverges at a typical point. By a 'typical martingale', we mean an element of a co-$\sigma$-porous set when $p\in[1,\infty)$ and of a co-porous set when $p=\infty$. Specifically, we show that while a typical martingale diverges on a co-$\sigma$-porous set with respect to the first metric, no martingale diverges on a co-$\sigma$-porous set with respect to the second metric. Finally, we investigate the key factors determining whether the divergence set can be co-$\sigma$-porous.

We study the long-term distributional dynamics and small-noise asymptotics of two-dimensional nonautonomous stochastic primitive equations driven by Gaussian and multiplicative Lévy noise, together with moderate deviations for the purely jump model. For sufficiently small noise, uniform moment bounds and an exponentially weighted terminal estimate yield a pullback absorbing family and tightness, while a lower semicontinuous vertical-moment functional preserves admissibility under weak limits. We prove the existence and uniqueness of a pullback measure attractor in the weak topology of probability measures and establish its upper semicontinuity as both noise components vanish. For the jump-driven equation, we establish a moderate deviation principle in $\mathcal D([0,T];H)\cap L^2(0,T;V)$ with speed $a^2(\epsilon)/\epsilon$. The proof combines continuity of the skeleton map with controlled stochastic convergence based on entropy bounds, truncation, martingale estimates, and direct vertical estimates, avoiding Lipschitz continuity of the vertical derivative of the jump coefficient.

We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised $(2,2)$-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Plücker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.

We consider a regularly varying stationary sequence of random variables (Xt) with tail index ___ < 2. For these sequences we study the joint convergence of sums, `p- type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes (Xt) whose limit ratios possess all moments. But, in contrast to the latter situation, there also exist sequences (Xt) where certain moments of the limit ratio are in___nite. This phenomenon results from extremal clusters in the sequence.

This paper addresses a class of integral representations of the form \begin{equation} f(t,x)=g(t,x)+\int_0^t k(t,s)\, p(t-s,x,y)\, \partial_x f(s,y^+)\,ds, \qquad 0 \le t \le T, \end{equation} where $f$ is unknown, $p$ is the transition density of a diffusion process, and $g, k$ are prescribed functions. For an arbitrary diffusion process with sufficiently regular coefficients, we prove that this problem is equivalent to a Volterra integral equation of the second kind. This reduction provides a unified framework for both theoretical analysis and numerical approximation. An example of the implementation in the context of financial mathematics is presented.

Let X be a spectrally negative Lévy process observed up to an inde- pendent exponential time T with parameter {\gamma} > 0. We study the joint law of the maximum drawdown and the maximum drawup of X on [0, T ]. The path is decomposed according to the two possible orderings of its in- fimum and supremum. Conditional on the values of these extrema and on their ordering, the resulting pre-, intermediate, and post- components are independent. We identify their laws as Doob h-transforms of killed spec- trally negative Lévy processes and express the corresponding distribution functions explicitly in terms of the {\gamma}-scale functions W ({\gamma}) and Z({\gamma}) and their derivatives. Combining the conditional laws with the joint densities of the extrema yields integral representations of the joint distribution and moments.

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $\eta\in(0,\infty)$ and ResNets $\Psi_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $\eta d^{\eta}\varepsilon^{-\eta}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $\xi>0$, the explicit bound $C_\xi d^{4+\xi}\varepsilon^{-(3+\xi)}$ on the number of parameters.

In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. We show that, as $N\to\infty$, the boundary of the complex eigenvalue distribution of $\mathbf{A}\mathbf{B}$ is governed by simple equations involving the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$.