Guangqian Zhao
We study a two-component fractional stochastic Klein--Gordon system on $\mathbb{T}^3$ driven by independent space-time white noises. Our main result is the identification of a new color-speed separation principle for the mixed paracontrolled random operators in the multispeed regime. This principle is created by the interaction between probabilistic cancellation, multispeed dispersive phase separation, and the nonlinear random-operator structure generated by the paracontrolled expansion. At the level of the nonlinear expansion, it reorganizes the singular contraction geometry of the mixed random operators: cross-color contraction channels vanish by independence, while the surviving same-color contractions are forced into speed-separated oscillatory interactions and therefore become finite Volterra-type deterministic multipliers. In this way, the multispeed structure does not merely alter the linear phase geometry; it changes the nonlinear singular interaction mechanism itself and regularizes the most singular contraction channels produced by the quadratic coupling. We prove the almost sure pathwise convergence of the centered fluctuation operators and the convergence of the same-color deterministic contractions, with no singular random-operator centering required in the limiting equation. Our analysis identifies $α= \frac{3}{4}$ as the intrinsic structural threshold of the mixed random-operator algebra. In addition, we construct the baseline stochastic lift and formulate a conditional deterministic closure for the full paracontrolled system. These results show that, in coupled singular dispersive SPDEs, multispeed propagation can act as a geometric regularizer precisely through its effect on the nonlinear random-operator structure.
Yuli Wu
This note provides a chronological account of Fréchet distances, starting with Maurice Fréchet's 1906 doctoral thesis on distances in abstract sets and tracing the Fréchet distance between polygonal curves and its algorithmic computation in the 1990s. It then continues with his 1957 paper on a coupling-based distance between probability laws with a brief glimpse of Wasserstein distance and optimal transport. We further attempt to draw connections between the distributional, coupling-based facet of Fréchet distances on probability laws and the geometric facet on curves. The note ends with a modern use case, the Fréchet Inception Distance (FID) in the era of deep generative model evaluation, interpretable as the Wasserstein-2 distance between multivariate Gaussians in a learned feature space. An appendix includes \TeX{}ified faithful English translations of Fréchet's 1906 thesis and 1957 paper, and Lévy's 1950 note for reader convenience.
Nikeethan Selvaratnam, Dorinel Bastide, Clément Fernandes, Wojciech Pieczynski
Apr 23, 2026·q-fin.RM·PDF Predicting future operational risk losses gives rise to a significant challenge due to the heterogeneous and time-dependent structures present in real-world data. Furthermore, stress test exercises require examining the relationship with operational losses. To capture such relationship, we propose to use an extension of Hidden Markov Models to multivariate observations. This model introduces a third auxiliary variable designed to accommodate the economic covariates in the time-series data. We detail the unique aspects of operational risk data and describe how model calibration is achieved via the Expectation-Maximization (EM) algorithm. Additionally, we provide the calibration results for the various risk-event types and analyze the relevance of the inclusion of the macroeconomic covariates.
Moritz Dober, Alexander Glazman, Sébastien Ott
The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point $T_c(q)$, there are exactly $q+1$ extremal Gibbs measures (pure phases): $q$ ordered (monochromatic) and one disordered (free). This work establishes for the first time the wetting phenomenon in a precise geometric form and in the entire regime of discontinuity $q>4$: at $T_c(q)$, between two ordered phases a disordered layer emerges and, in the diffusive scaling, its boundaries converge to a pair of Brownian motions conditioned not to intersect. This is starkly different from the subcritical ($T<T_c(q)$) behaviour. At $T_c(q)$, previous results (Bricmont--Lebowitz '87, Messager--Miracle-Sole--Ruiz--Shlosman '91) were limited to the construction and properties of the surface tension for large enough $q$. In a companion work, arXiv:2502.04129, we provide a detailed study of the Potts model under order-disorder Dobrushin conditions. That work also develops a ``renewal picture'' à la Ornstein-Zernike for a suitable percolation model, which plays a central part in our study of the Potts interfaces. The latter is the random-cluster representation of an Ashkin--Teller model (ATRC), and is related to the Potts model via a chain of couplings going through the six-vertex model. In the current work, we extend the analysis to a pair of interacting order-disorder interfaces forming the separation between the two ordered phases, and couple them to a pair of well-behaved random walks conditioned not to intersect. The construction of the coupling is based on rigorously deriving entropic repulsion between the two interfaces. We also prove convergence of interfaces in the FK-percolation model at $p_c(q)$ when $q>4$.
Léo Daures
We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely self-contained and relies on subadditivity. In the absence of irreducibility, examples show that the rate function is not convex in general.
Jiangwei Zhang
This paper investigates the long-time dynamics of solutions for an abstract nonlinear stochastic hydrodynamic-type equation driven by multiplicative Lévy noise. The framework encompasses several key hydrodynamical models, including the stochastic 2D Navier-Stokes equations, magnetohydrodynamic equations, the magnetic Bérnard problem, as well as various stochastic shell models of turbulence. Under the assumption that the nonlinear noise coefficients satisfy local Lipschitz and linear growth conditions, we first establish global well-posedness using a truncation technique. Then, by introducing a mean random dynamical system, we prove the existence and uniqueness of weak pullback mean random attractors for the system. Furthermore, when the external force is time-independent, we study the existence of invariant measures for the corresponding autonomous system, as well as the double limiting behavior of invariant measures with respect to the intensities of Gaussian and Lévy noise. Finally, under additional assumptions on the bilinear nonlinear term (e.g., as in the Navier-Stokes equations), we examine the existence and uniqueness of pullback measure attractors, along with the asymptotically autonomous stability of such attractors as the time parameter tends to negative infinity. It is worth noting that the results of this paper are new even for the single stochastic 2D Navier-Stokes equations.
Phan Thành Nam, Rongchan Zhu, Xiangchan Zhu
We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure.
Marcel Nutz, Chenyang Zhong
We study the vanishing-regularization limit of entropically regularized optimal transport (EOT) for the Euclidean distance cost $c(x,y)=\|x-y\|$ in dimension $d>1$. We develop a comprehensive variational convergence framework that entails two main results. First, we resolve the longstanding entropic selection problem: the EOT minimizer converges to a distinguished optimal transport plan that is characterized explicitly as the solution of a constrained EOT problem on each transport ray. Denoting by $\varepsilon>0$ the regularization parameter, this selection holds for all $o(\varepsilon)$-approximate minimizers, with sharp failure at the $O(\varepsilon)$ scale. Second, we establish an explicit second-order expansion of the entropic transport cost. The second-order term encodes the geometry of the regularization and reveals the optimal asymptotic tradeoff between entropy and transport cost.
Gianlorenzo D'Angelo, Riccardo Michielan
We study the efficient generation of random graphs with a prescribed expected degree sequence, focusing on rank-1 inhomogeneous models in which vertices are assigned weights and edges are drawn independently with probabilities proportional to the product of endpoint weights. We adopt a temporal viewpoint, adding edges to the graph one at a time up to a fixed time horizon, and allowing for self-loops or duplicate edges in the first stage. Then, the simple projection of the resulting multigraph recovers exactly the simple Norros--Reittu random graph, whose expected degrees match the prescribed targets under mild conditions. Building on this representation, we develop an exact generator based on \textit{edge-arrivals} for expected-degree random graphs with running time $O(n+m)$, where $m$ is the number of generated edges, and hence proportional to the output size. This removes the typical vertex sorting used by widely-used fast generator algorithms based on \textit{edge-skipping} for rank-1 expected-degree models, which leads to a total running time of $O(n \log n + m)$. In addition, our algorithm is simpler than those in the literature, easy to implement, and very flexible, thus opening up to extensions to directed and temporal random graphs, generalization to higher-order structures, and improvements through parallelization.
Sara Mazzonetto, Benoît Nieto
We study existence and uniqueness for one-dimensional generalized stochastic differential equations with singular coefficients, including distributional drift and degenerate, possibly discontinuous, diffusion coefficients. Such singularities naturally encode changes in the dynamics at thresholds, including reflecting, skew, or sticky interface behavior. We develop two directions. We provide sufficient conditions for pathwise uniqueness, under weak existence and uniqueness in law, without assuming uniform ellipticity or continuity of the diffusion coefficient. We also investigate a pasting approach for generalized stochastic differential equations that transfers strong existence and pathwise uniqueness, as well as weak existence and uniqueness in law, from local component equations to a global solution. To the best of our knowledge, this provides the first explicit pasting theorem yielding pathwise uniqueness in the setting of generalized stochastic differential equations. As an application, we establish the first existence and uniqueness results for a class of skew sticky threshold Cox-Ingersoll-Ross-type diffusions, including the threshold Chan-Karolyi-Longstaff-Sanders process.
David Vernotte
We study some geometric properties of the excursion set of a slope field alpha associated to a smooth, planar, centered, Gaussian field f. That, is we consider the set of all points such that the value of alpha is at most l where l is a real parameter called the level. We restrict our attention to the levels l that are supercritical. We show that for almost such l, in the sense of the Lebesgue measure, then with high probability the chemical distance between two points connected in the excursion set at level l is comparable to the usual Euclidean distance between those two points. This result is in the spirit of the Antal Pisztora theorem for Bernoulli percolation. However, many new difficulties arise such as the fact that alpha is a continuous field (not differentiable everywhere) with long range correlations and whose law is still not well understood.
Jiang Zhou, Ziru Deng, Pengcheng Hou
We present a Monte Carlo study of the fractal geometry of clusters formed by discrete-time simple random walks (sRW) of $L^2$ steps on a periodic square $L\times L$ lattice. We verify with high precision that the asymptotic behavior of the cluster mass follows $M/L^2 \simeq (\ln L)^{-1} [\fracπ{2}+b (\ln L)^{-2}]$, with $b\approx -(π/2)^{-2}$, demonstrating marginal ``logarithmic fractals". We further determine the fractal dimension of the hull to be $d_{\rm hull}=1.333\,29(14)=4/3$, in excellent agreement with the prediction of Schramm-Loewner evolution ($\rm SLE_{8/3}$) for the Brownian frontier universality class. More importantly, we analyze the chemical distance $S$ spanning the cluster and obtain strong evidence that it asymptotically scales as $S\sim L(\ln L)^{1/4}$, lying exactly on the theoretical upper bound for the chemical distance for level-set percolation clusters on the two-dimensional Gaussian free field. Our numerical results show that the sRW cluster exhibits a conformally invariant external frontier and contains highly efficient asymptotically linear connective paths.
Xin Chen, Chenlin Gu, Jian Wang
In this paper, we study the stochastic homogenization for a class of symmetric random walks in random conductance model, whose one-step transition probability from $x$ to $y$ is proportional to $|x-y|^{-d-2}$. As the associated jumping kernel fails to be $L^2$-integrable yet admits a finite $α$-th moment for all $α\in (0,2)$, we refer to the corresponding process $(X^\w_t)_{t\ge0}$ as a long-range random walk with critical jump index. In this critical regime, the scaled process $\bigl(k^{-1}X_{k^2(\log k)^{-1}t}\bigr)_{t\ge 0}$, whose scaling order is different from the diffusive scaling and the $α$-stable scaling, converges to a Brownian motion. Besides characterizing the limiting Brownian motion, we will give a convergence rate for associated scaled resolvents, which obeys the order $(\log k)^{-\frac{1}{2}+\frac{1}{2(d-2)}+\varepsilon}$ with any $\varepsilon>0$ for all $d>3$.
Ahmed Bou-Rabee, Paul Dario
We consider the long-range random conductance model on $\mathbb{Z}^d$ at the critical exponent: the jump rate between sites $x$ and $y$ decays as $\mathbf{a}(x,y) |x-y|^{-(d+2)}$, where $\mathbf{a}(x,y)$ are i.i.d. uniformly elliptic conductances. Below the critical exponent $(d+2)$ the walk converges to a stable process; above it, to Brownian motion with diffusive $\sqrt{t}$ scaling. At criticality the second moment of the jump kernel diverges logarithmically. We establish quantitative homogenization of the associated elliptic equation to the Laplacian at the rate $1/\sqrt{|\ln\varepsilon|}$. As a consequence, we deduce quenched convergence of the random walk to Brownian motion under the anomalous $\sqrt{t \log t}$ scaling. Unlike in standard homogenization, the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order.
Ahmed Bou-Rabee, Vittoria Silvestri, Ariel Yadin
Internal DLA is a discrete random growth model describing growing clusters of particles. Its limiting shape and fluctuations are well understood when the underlying graph is the $d$-dimensional lattice or the cylinder $\mathbb{Z}_N \times \mathbb{Z}$. In the latter geometry, the average fluctuations of IDLA have been shown to converge to the GFF. In this note we generalise this result by showing that, for any vertex-transitive base graph $V_N$ satisfying an eigenvalue convergence condition, the average fluctuations of IDLA on the cylinder $V_N \times \mathbb{Z}$ are given by a GFF. On the way, we present an improved bound on the clusters' maximal fluctuations, which is of independent interest and which implies a shape theorem for IDLA on $V_N \times \mathbb{Z}$ for any vertex-transitive base graph $V_N$.
Stephen Jordan Harrison
We derive upper bounds for probabilities of the form $P(g(\mathbf{X})\ge t)$ using the southwest boundary (recently introduced in our previous work) $\partial_{\mathrm{SW}} Q(g^{-1}[t,\infty))$, where $Q$ is a reflection to the first quadrant. Under natural continuity, symmetry, and monotonicity assumptions on $g$, this yields explicit and computable bounds of the form $P(g(\mathbf{X})\ge t)\le ns_t$, where $s_t$ is the unique parameter at which the line $L(s)=(f_1^{-1}(s),\dots,f_n^{-1}(s))$ intersects the southwest boundary. In particular, when $g$ is a homogeneous polynomial of degree $k$ (plus a constant $C$) with positive coefficients and all tail bounds on the random variables are identical, the bound simplifies to the closed-form expression $$ P(g(\mathbf{X})\ge t)\le nf\bigg(\frac{(t-C)^{1/k}}{(\sum_i|a_i|)^{1/k}}\bigg) $$ where $a_i$ are the coefficients on the monomials in $g$. We then obtain an explicit tail bound for the trace of a Schur multiplier acting on random matrices with identical tail bounds on the random variables. No assumptions are made about independence or dependence.
A. Mazel, I. Stuhl, Y. Suhov
We prove that the maximal graph-density of an independent set in a Penrose P3 tiling considered as a planar non-directed graph is equal to $(57 - 25 \sqrt{5})/2 \approx 0.54915$ despite the fact that the graph is bipartite. Accordingly, the extreme Gibbs measure of the nearest-neighbor hard core particle model on this graph is unique for sufficiently large values of the particle activity. This invalidates a natural expectation to observe the coexistence of even and odd phases.
Andrea Kubin, Enrico Pasqualetto
We develop a measure and integration theory for random normed modules. Given a probability space $({\rm X},Σ,\mathfrak m)$, we introduce and study measures taking values into the space $L^0(\mathfrak m)$ of $\mathfrak m$-measurable functions quotiented up to $\mathfrak m$-a.e. equality. Moreover, we develop a Bochner-type integration theory with respect to an $L^0(\mathfrak m)$-valued measure $μ$, for maps whose target ${\rm M}$ is a complete random normed module with base $({\rm X},Σ,\mathfrak m)$, or equivalently an $L^0(\mathfrak m)$-Banach $L^0(\mathfrak m)$-module. Inter alia, we prove versions of the Radon-Nikodým theorem and of the Riesz-Markov-Kakutani representation theorem for $L^0(\mathfrak m)$-valued measures. We also outline several applications of our integration theory: we introduce a notion of martingale with values in a complete random normed module, we propose a definition of random Radon-Nikodým property and we discuss random sets of finite perimeter.
Edith J. Zhang, James Scott, Qiang Du
A graph reaction--diffusion (RD) equation is a system of differential equations that is defined on the nodes of a graph. Consider a sequence of growing graphs that converges in cut norm to a limiting graphon. We show that the solutions of the sequence of graph RD equations converge in $L^p$ norm, for $p \in [1,\infty]$, to the solution of a limiting nonlocal RD equation, which we call a graphon RD equation. Furthermore, we show a large numbers result for a stochastic particle process that consists of a random walk and a birth-death process on graphs. For a sequence of graphs that converge in cut norm to a limiting graphon, the sequence of stochastic processes converges in probability to the solution of the graphon RD equation.
Maja Gwozdz
Large-deviation upper bounds on compact sets do not, in general, extend to arbitrary closed sets without additional tightness. We show that this obstruction already occurs in static entropic optimal transport. More precisely, we construct a fixed-cost model with continuous cost and nonatomic marginals for which the entropic minimisers converge in total variation to an optimal plan with noncompact support, the known compact-set upper bound remains valid, but the corresponding closed-set upper bound fails on a specific closed subset of the ambient space. For a fixed closed set, we identify the exact tail criterion for passing from compact to closed sets. We show that there does not exist a full large-deviation principle (LDP) on the ambient space at speed $1/\varepsilon$ with an arbitrary lower semicontinuous rate function.