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新提交 (展示 7 之 7 条目 )

[1] arXiv:2508.04089 [中文pdf, pdf, 其他]

标题： 长时间行为和实值性状结构的出生与死亡过程的Yaglom极限 显示英文标题

标题： Long time behavior and Yaglom limit for real trait-structured Birth and Death Processes

Pierre Collet (CPHT), Sylvie Méléard (MERGE, CMAP), Jaime San (CMM)

主题： 概率 (math.PR)

在本文中，我们研究连续时间中测度值的出生与死亡过程的长期行为，其中跳跃之间的动力学是一维马尔可夫过程，包括扩散和跳跃。 我们考虑三种情形，临界、次临界和超临界。 在对费曼-卡茨半群的适当假设下，我们证明了矩和灭绝概率的新遍历性，它们的时间渐进行为以及在不灭绝条件下的测度值出生与死亡过程的依分布收敛，从而导致Q过程和亚格龙极限（在无限维设置中）的存在性。 我们开发了三类自然例子，我们的结果适用于这些例子。 显示英文摘要

In this article we study the long time behaviour of measure-valued birth and death processes in continuous time, where the dynamics between jumps are one-dimensional Markov processes including diffusion and jumps. We consider the three regimes, critical, subcritical and supercritical. Under suitable hypotheses on the Feynman-Kac semigroup, we prove a new recurrence for the moments and the extinction probability, their time asymptotics and the convergence in law for the measure-valued birth and death process conditioned to non extinction, leading to the existence of Q-process and Yaglom limit (in this infinite dimensional setting). We develop three classes of natural examples where our results apply.

[2] arXiv:2508.04327 [中文pdf, pdf, html, 其他]

标题： 矩阵罗森塔尔不等式和马尔可夫链的集中不等式及其在统计学习中的应用 显示英文标题

标题： Matrix Rosenthal and Concentration Inequalities for Markov Chains with Applications in Statistical Learning

Yang Peng, Yuchen Xin, Zhihua Zhang

主题： 概率 (math.PR) ; 统计理论 (math.ST)

在本文中，我们研究了依赖随机矩阵和的矩和浓度不等式。我们建立了矩阵鞅的新Rosenthal-Burkholder不等式，以及针对遍历马尔可夫链的矩阵Rosenthal、Hoeffding和Bernstein不等式。与之前关于马尔可夫链矩阵浓度不等式的研究相比，我们的结果不需要非零绝对谱间隙和有界矩阵函数的假设。此外，我们的结果的主要项符合马尔可夫链中心极限定理，而不是依赖于方差代理。我们还给出了与环境维度$d$无关的不等式版本，而是依赖于某些矩阵的有效秩。这使得我们的结果可以推广到无限维希尔伯特空间中的线性算子。我们的结果在统计学和机器学习中有广泛的应用；特别是，我们在马尔可夫数据上的协方差估计和主成分分析中获得了改进的界限。 显示英文摘要

In this paper, we study moment and concentration inequalities of the spectral norm for sums of dependent random matrices. We establish novel Rosenthal-Burkholder inequalities for matrix martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities for ergodic Markov chains. Compared with previous work on matrix concentration inequalities for Markov chains, our results do not require the assumptions of a non-zero absolute spectral gap and bounded matrix functions. Furthermore, our results have leading terms that match the Markov chain central limit theorem, rather than relying on variance proxies. We also give dimension-free versions of the inequalities, which are independent of the ambient dimension $d$ and relies on the effective rank of some matrix instead. This enables the generalization of our results to linear operators in infinite-dimensional Hilbert spaces. Our results have extensive applications in statistics and machine learning; in particular, we obtain improved bounds in covariance estimation and principal component analysis on Markovian data.

[3] arXiv:2508.04373 [中文pdf, pdf, html, 其他]

标题： 微分香农和Rényi熵的重新审视 显示英文标题

标题： Differential Shannon and Rényi entropies revisited

Yuliya Mishura, Kostiantyn Ralchenko

评论： 18页，5图

主题： 概率 (math.PR)

香农熵对于离散分布是一个基本且广泛使用的概念，但其连续类比，称为微分熵，缺乏一些基本性质，如正性和与离散情况的兼容性。 在本文中，我们详细分析了这种不兼容性并通过示例进行说明。 为了克服这些限制，我们提出了改进的香农熵和瑞尼熵版本，它们保留了关键性质，包括正性，同时仍接近经典形式。 我们还定义了兼容的离散泛函，并研究了所提出的熵对于正态分布和指数分布的行为。 显示英文摘要

Shannon entropy for discrete distributions is a fundamental and widely used concept, but its continuous analogue, known as differential entropy, lacks essential properties such as positivity and compatibility with the discrete case. In this paper, we analyze this incompatibility in detail and illustrate it through examples. To overcome these limitations, we propose modified versions of Shannon and R\'enyi entropy that retain key properties, including positivity, while remaining close to the classical forms. We also define compatible discrete functionals and study the behavior of the proposed entropies for the normal and exponential distributions.

[4] arXiv:2508.04386 [中文pdf, pdf, html, 其他]

标题： 随机正规矩阵计数统计波动的普遍性 显示英文标题

标题： Universality for fluctuations of counting statistics of random normal matrices

J. Marzo, L. D. Molag, J. Ortega-Cerdà

评论： 31页

主题： 概率 (math.PR) ; 数学物理 (math-ph)

我们考虑依赖于势函数$Q$的$n\times n$随机正规矩阵在给定集合$A$中的特征值数量的波动。 这些特征值已知形成一个行列式点过程，并且在对$Q$有温和条件的情况下，它们会聚集在一个称为液滴的紧集上。 当$A$是滴落内部的 Borel 集时，我们证明了在$A$中特征值$N_A^{(n)}$的方差具有由以下给出的极限行为 \begin{align*} \lim_{n\to\infty} \frac1{\sqrt n}\operatorname{Var } N_A^{(n)} = \frac{1}{2\pi\sqrt\pi}\int_{\partial_* A} \sqrt{\Delta Q(z)} \, d\mathcal H^1(z), \end{align*}其中$\partial_* A$是$A$的测度论边界，$d\mathcal H^1(z)$表示一维豪斯多夫测度，而$\Delta = \partial_z \overline{\partial_z}$。 我们也考虑当$A$是液滴的微观扩张的情况，并对 Akemann, Byun 和 Ebke 的结果进行完全推广，适用于任意势能。在这个结果中，$d\mathcal H^1(z)$被替换为与液滴外部相关的在$\infty$处的调和测度。这个第二个结果是通过加强 Hedenmalm-Wennman 和 Ameur-Cronvall 关于相关关联核在液滴边界附近的渐近行为的结果来证明的。 显示英文摘要

We consider the fluctuations of the number of eigenvalues of $n\times n$ random normal matrices depending on a potential $Q$ in a given set $A$. These eigenvalues are known to form a determinantal point process, and are known to accumulate on a compact set called the droplet under mild conditions on $Q$. When $A$ is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues $N_A^{(n)}$ in $A$ has a limiting behavior given by \begin{align*} \lim_{n\to\infty} \frac1{\sqrt n}\operatorname{Var } N_A^{(n)} = \frac{1}{2\pi\sqrt\pi}\int_{\partial_* A} \sqrt{\Delta Q(z)} \, d\mathcal H^1(z), \end{align*} where $\partial_* A$ is the measure theoretic boundary of $A$, $d\mathcal H^1(z)$ denotes the one-dimensional Hausdorff measure, and $\Delta = \partial_z \overline{\partial_z}$. We also consider the case where $A$ is a microscopic dilation of the droplet and fully generalize a result by Akemann, Byun and Ebke for arbitrary potentials. In this result $d\mathcal H^1(z)$ is replaced by the harmonic measure at $\infty$ associated with the exterior of the droplet. This second result is proved by strengthening results due to Hedenmalm-Wennman and Ameur-Cronvall on the asymptotic behavior of the associated correlation kernel near the droplet boundary.

[5] arXiv:2508.04501 [中文pdf, pdf, html, 其他]

标题： 动态在无序能量景观上的局部-全局相关性 显示英文标题

标题： Local-global correlations of dynamics on disordered energy landscapes

Jacob Calvert, Dana Randall

评论： 17页，3图

主题： 概率 (math.PR) ; 统计力学 (cond-mat.stat-mech)

连续时间马尔可夫链的平稳分布通常源于概率通量的复杂全局平衡。 然而，实证证据表明，定义为平稳分布的负对数的有效势能，通常与状态的一个简单局部属性：其逃逸率的对数，高度相关。 为了更好地理解为什么以及如何通常这种相关性很高，我们研究了在由高斯势阱和势垒组成的能量景观上的可逆反应动力学，这些势阱和势垒分别与正则图的顶点和边相关。 我们发现，只要势垒的高度变化显著小于势阱的深度，无论底层图的度数如何，这种相关性就足够高。 作为应用，我们给出了随机能量模型动力学所表现出的期望相关性的下界，该模型被称为Bouchaud陷阱模型。 我们预计，该证明结合了期望相关性的通用下界与高斯浓度不等式，可以扩展到几个其他模型类别。 显示英文摘要

The stationary distribution of a continuous-time Markov chain generally arises from a complicated global balance of probability fluxes. Nevertheless, empirical evidence shows that the effective potential, defined as the negative logarithm of the stationary distribution, is often highly correlated with a simple local property of a state: the logarithm of its exit rate. To better understand why and how typically this correlation is high, we study reversible reaction kinetics on energy landscapes consisting of Gaussian wells and barriers, respectively associated with the vertices and edges of regular graphs. We find that for the correlation to be high it suffices for the heights of the barriers to vary significantly less than the depths of the wells, regardless of the degree of the underlying graph. As an application, we bound below the expected correlation exhibited by dynamics of the random energy model, known as the Bouchaud trap model. We anticipate that the proof, which combines a general lower bound of the expected correlation with the Gaussian concentration inequality, can be extended to several other classes of models.

[6] arXiv:2508.04637 [中文pdf, pdf, html, 其他]

标题： 耦合KPZ方程及其可解耦性 显示英文标题

标题： Coupled KPZ equations and their decoupleability

Boliang Fu, Tadahisa Funaki, Sunder Sethuraman, Shankar Venkataramani

评论： 31页

主题： 概率 (math.PR) ; 代数几何 (math.AG)

我们讨论三线性或完全对称的实数$n\times n\times n$张量的解耦性，包括部分和完全解耦性，这些信息揭示了某些耦合KPZ方程的结构。 非正式地说，当张量是部分可解耦时，耦合KPZ方程中的一个分量会从其他分量中分离出来，而当张量是完全可解耦时，每个$n$分量都会从其他分量中分离出来。 这种特征被重新表述为三线性张量属于$O(n)$轨道的问题，这些轨道是完全可解耦和部分可解耦张量的子集。 当$n=2$时，这些子集是相同的，并且在这种情况下，给出一个关于张量条目的一般条件来判断是否属于这些子集的轨道。 当$n\geq 3$时，这些子集是不同的。 对于$n\geq 3$，我们用几个抽象关系来描述完全解耦性，当$n=3$时，这些关系变得明确。 当$n=3$时，我们还明确地描述了部分解耦性。 这些方法涉及应用不变量理论中的概念，将$O(n)$不变子集与在更小集合上的稳定子群作用联系起来。 当$n=3$使用 Olive 和 Auffray 找到的不变量显式基时。 当$n=2$时，我们还提供了另外两种更直接的论证。 显示英文摘要

We discuss characterizations of the decoupleability, partial and full, of trilinear or completely symmetric real $n\times n\times n$ tensors, which inform on the structure of certain coupled KPZ equations. Informally, when the tensor is partially decoupleable, one of the components in the coupled KPZ equation splits off from the others, while when the tensor is fully decoupleable, each of the $n$ components splits off from the others. Such a characterization is recast as a problem of membership of trilinear tensors in $O(n)$ orbits of subsets of fully decoupleable and partially decoupleable tensors. When $n=2$, we show these subsets are the same, and in this case give a single criterion in terms of the entries of a tensor for membership in the orbits of these subsets. When $n\geq 3$, the subsets are different. For $n\geq 3$, we characterize full decoupleability in terms of several abstract relations, which when $n=3$ are made explicit. When $n=3$, we also explicitly characterize partial decoupleability. The methods involve notions in applied invariant theory, relating $O(n)$ invariant subsets to stabilizer subgroup actions on smaller sets. When $n=3$ make use of the explicit basis of invariants found by Olive and Auffray. When $n=2$, we also supply two other more direct arguments.

[7] arXiv:2508.04640 [中文pdf, pdf, html, 其他]

标题： 在$\mathbb{R}^d$中的布丰-拉普拉斯针问题中交叉次数的概率 显示英文标题

标题： The probabilities for the number of intersections in the Buffon-Laplace needle problem in $\mathbb{R}^d$

Uwe Bäsel

评论： 12页，2图

主题： 概率 (math.PR)

1974年，Stoka在$\mathbb{R}^d$，$d \ge 2$中解决了布丰的针的问题，即他找到了一个闭式解，用于计算长度为$\ell$的线段（“针”）与间距为$a\ge\ell$的平行超平面网格相交的概率。 对于 $\mathbb{R}^d$中的拉普拉斯针问题，其中存在 $d$组平行超平面，间距为 $a_1,\ldots,a_d$，满足 $\min(a_1,\ldots,a_d)\ge\ell$，且法向量方向为坐标轴方向 $x_1,\ldots,x_d$，他只能给出针同时与所有组的超平面相交的情况的闭合解。 在本文中，我们计算了针与由 $d$ 家族形成的超矩形网格之间的恰好 $i$， $0\le i\le d$，交叉点的概率 $p_d(i)$，并得出了交叉点数量的期望值和方差。 此外，我们提出一个模拟程序和一些数值结果。 显示英文摘要

In 1974, Stoka solved Buffon's needle problem in $\mathbb{R}^d$, $d \ge 2$, i.e. he found a closed form solution for the probability that a line segment ("needle") with length $\ell$ intersects a grid of parallel hyperplanes with mutual distance $a\ge\ell$. For the Laplace needle problem in $\mathbb{R}^d$, where there are $d$ families of parallel hyperplanes with distances $a_1,\ldots,a_d$ fulfilling $\min(a_1,\ldots,a_d)\ge\ell$, and normal vectors in the direction of the coordinate axes $x_1,\ldots,x_d$, he was only able to give a closed solution for the case that the needle intersects hyperplanes of all families simultaneously. In the present paper, we calculate the probabilities $p_d(i)$ of exactly $i$, $0\le i\le d$, intersection points between the needle and the hyperrectangular grid formed by the $d$ families, and conclude the expected value and the variance for the number of intersection points. Furthermore, we present a simulation program and some numerical results.

交叉提交 (展示 10 之 10 条目 )

[8] arXiv:2508.03790 (交叉列表自 math.ST) [中文pdf, pdf, html, 其他]

标题： 正态分布的渐近普遍矩匹配性质 显示英文标题

标题： Asymptotic universal moment matching properties of normal distributions

Xuan Liu

主题： 统计理论 (math.ST) ; 概率 (math.PR) ; 计算金融 (q-fin.CP)

矩匹配是一种易于实现且通常有效的减少蒙特卡罗模拟估计方差的方法。 另一方面，对于一般的积分问题，不能保证矩匹配在样本数量较大时至少渐近地减少模拟方差，即： 当样本数量很大时。 我们研究了在给定的基本分布$X$下，对于一般的积分问题$\mathbb{E}[f(X)]$，应用矩匹配技术时保证渐近方差减少的条件的特征。 我们证明，这种渐近方差减少性质的充分必要条件是$X$是正态分布。 此外，当$X$是正态分布时，得到了用于有效估计（一阶和二阶）矩匹配蒙特卡罗模拟方差的公式。 这些公式允许在模拟过程中以类似于普通蒙特卡罗方差估计的方式，作为模拟过程的副产品来估计模拟方差。 此外，我们提出了针对任何给定连续分布的非线性矩匹配方案，以保证渐近方差减少。 显示英文摘要

Moment matching is an easy-to-implement and usually effective method to reduce variance of Monte Carlo simulation estimates. On the other hand, there is no guarantee that moment matching will always reduce simulation variance for general integration problems at least asymptotically, i.e. when the number of samples is large. We study the characterization of conditions on a given underlying distribution $X$ under which asymptotic variance reduction is guaranteed for a general integration problem $\mathbb{E}[f(X)]$ when moment matching techniques are applied. We show that a sufficient and necessary condition for such asymptotic variance reduction property is $X$ being a normal distribution. Moreover, when $X$ is a normal distribution, formulae for efficient estimation of simulation variance for (first and second order) moment matching Monte Carlo are obtained. These formulae allow estimations of simulation variance as by-products of the simulation process, in a way similar to variance estimations for plain Monte Carlo. Moreover, we propose non-linear moment matching schemes for any given continuous distribution such that asymptotic variance reduction is guaranteed.

[9] arXiv:2508.03833 (交叉列表自 math.ST) [中文pdf, pdf, html, 其他]

标题： 可计算的强逼近界及其应用 显示英文标题

标题： Computable Bounds for Strong Approximations with Applications

Haoyu Ye, Morgane Austern

主题： 统计理论 (math.ST) ; 概率 (math.PR)

Komlós$\unicode{x2013}$Major$\unicode{x2013}$Tusnády (KMT) 不等式对于部分和而言是概率论中最著名的成果之一。 然而，由于其依赖于未知常数，因此在实际应用中受到阻碍。 本文针对有界独立同分布随机变量解决了这一限制。 在付出额外的对数因子代价的情况下，我们提出了一种可计算的 KMT 不等式版本，该版本仅依赖于变量的范围和标准差。 我们还推导出了一个经验性的不等式版本，在标准差未知时也能实现名义覆盖率。 随后，我们通过在线变化点检测和首次到达时间概率的应用展示了我们界限的实用性。 显示英文摘要

The Koml\'os$\unicode{x2013}$Major$\unicode{x2013}$Tusn\'ady (KMT) inequality for partial sums is one of the most celebrated results in probability theory. Yet its practical application is hindered by its dependence on unknown constants. This paper addresses this limitation for bounded i.i.d. random variables. At the cost of an additional logarithmic factor, we propose a computable version of the KMT inequality that depends only on the variables' range and standard deviation. We also derive an empirical version of the inequality that achieves nominal coverage even when the standard deviation is unknown. We then demonstrate the practicality of our bounds through applications to online change point detection and first hitting time probabilities.

[10] arXiv:2508.03853 (交叉列表自 math.MG) [中文pdf, pdf, html, 其他]

标题： 单位酉不变矩阵范数的分离模数 显示英文标题

标题： The separation modulus of unitarily invariant matrix norms

Mustafa Alper Gunes, Assaf Naor

主题： 度量几何 (math.MG) ; 泛函分析 (math.FA) ; 概率 (math.PR) ; 谱理论 (math.SP)

如果 $X=(M_n(\mathbb{R}),\|\cdot\|)$ 是一个在 上的酉不变范数空间，那么我们证明（通过雅可比正交随机矩阵系综的精确计算）在单位球 $B_X$ 上带有狄利克雷边界条件的拉普拉斯算子的谱间隙 $X$ 满足 $\lambda(X)\asymp n^3 \|I\|^2$。 这导致了对$X$的弱同构逆等周猜想的确认，即我们证明存在一个凸体$L=L_X\subset B_X$，使得$\mathrm{vol}_{n^2}(L)^{1/n^2}\asymp \mathrm{vol}_{n^2}(B_{X})^{1/n^2}$，但其等周商至多是$n$的一个通用常数倍。 作为这些结果的推论（以及动机），我们得出$X$的分离模数满足$\mathsf{SEP}(X)\asymp \sqrt{n}\|I_n\|_X\mathrm{diam}(B_X)$，其中$\mathrm{diam}(B_X)$是$B_X$在$M_n(\mathbb{R})$上的标准欧几里得度量下的直径。 假设对$X$中的范数评估具有预言机访问权限，通过将其与一种新的确定性算法相结合，该算法可以高效地计算出由弱成员资格预言机给出且关于坐标排列和轴反射对称的凸体在$\mathbb{R}^n$中的$O(1)$-近似直径，我们得到一个预言机多项式时间算法，其输出保证在正的通用常数因子范围内是$X$的分离模量。 我们还推导出$X$的 Lipschitz 扩展模量的一个上界，在$X$为配备$\ell_{2}^n\to \ell_{2}^n$算子范数的$M_n(\mathbb{R})$的特殊情况下，该上界优于之前已知的最佳界限。 显示英文摘要

If $X=(M_n(\mathbb{R}),\|\cdot\|)$ is a unitarily invariant normed space on , then we prove (via exact computations for a Jacobi orthogonal random matrix ensemble) that the spectral gap of the Laplacian with Dirichlet boundary conditions on the unit ball $B_X$ of $X$ satisfies $\lambda(X)\asymp n^3 \|I\|^2$. This leads to a confirmation of the weak isomorphic reverse isoperimetry conjecture for $X$, namely, we demonstrate that there exists a convex body $L=L_X\subset B_X$ such that $\mathrm{vol}_{n^2}(L)^{1/n^2}\asymp \mathrm{vol}_{n^2}(B_{X})^{1/n^2}$, yet its isoperimetric quotient is at most a universal constant multiple of $n$. As a corollary (and motivation) of these results, we deduce that the separation modulus of $X$ satisfies $\mathsf{SEP}(X)\asymp \sqrt{n}\|I_n\|_X\mathrm{diam}(B_X)$, where $\mathrm{diam}(B_X)$ is the diameter of $B_X$ with respect to the standard Euclidean metric on $M_n(\mathbb{R})$. Assuming oracle access to norm evaluations in $X$, by combining this with a new deterministic algorithm for efficiently computing a $O(1)$-approximation of the diameter of convex bodies in $\mathbb{R}^n$ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the axes, we obtain an oracle polynomial time algorithm whose output is guaranteed to be the separation modulus of $X$ up to positive universal constant factors. We also deduce an upper bound on the Lipschitz extension modulus of $X$ that improves over the previously best-known bound even in the special case when $X$ is $M_n(\mathbb{R})$ equipped with the $\ell_{2}^n\to \ell_{2}^n$ operator norm.

[11] arXiv:2508.03961 (交叉列表自 math.CO) [中文pdf, pdf, html, 其他]

标题： 通过仿射谱独立性解耦：超越Banasczyk的Beck-Fiala和Komlós界限 显示英文标题

标题： Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk

Nikhil Bansal, Haotian Jiang

主题： 组合数学 (math.CO) ; 离散数学 (cs.DM) ; 数据结构与算法 (cs.DS) ; 概率 (math.PR)

贝克-菲拉猜想[离散应用数学，1981]断言，任何具有$n$个元素且度数为$k$的集合系统具有组合差异$O(\sqrt{k})$。一个重要的推广是科姆洛斯猜想，它指出任何具有单位长度列的$m \times n$矩阵具有差异$O(1)$。在本工作中，我们解决了$k \geq \log^2 n$的贝克-菲拉猜想。 We also give an $\widetilde{O}(\sqrt{k} + \sqrt{\log n})$ bound for $k \leq \log^2 n$, where $\widetilde{O}(\cdot)$ hides $\mathsf{poly}(\log \log n)$ factors. These bounds improve upon the $O(\sqrt{k \log n})$ bound due to Banaszczyk [Random Struct. Algor., 1998]. For the Komlos problem, we give an $\widetilde{O}(\log^{1/4} n)$ bound, improving upon the previous $O(\sqrt{\log n})$ bound [Random Struct. Algor., 1998]. All of our results also admit efficient polynomial-time algorithms. To obtain these results, we consider a new notion of affine spectral-independence in designing random walks. In particular, our algorithms obtain the desired colorings via a discrete Brownian motion, guided by a semidefinite program (SDP). Besides standard constraints used in prior works, we add some extra affine spectral-independence constraints, which effectively decouple the evolution of discrepancy across different rows, and allow us to better control how many rows accumulate large discrepancy at any point during the process. This technique of ``decoupling via affine spectral-independence'' is quite general and may be of independent interest. 显示英文摘要

The Beck-Fiala Conjecture [Discrete Appl. Math, 1981] asserts that any set system of $n$ elements with degree $k$ has combinatorial discrepancy $O(\sqrt{k})$. A substantial generalization is the Koml\'os Conjecture, which states that any $m \times n$ matrix with unit length columns has discrepancy $O(1)$. In this work, we resolve the Beck-Fiala Conjecture for $k \geq \log^2 n$. We also give an $\widetilde{O}(\sqrt{k} + \sqrt{\log n})$ bound for $k \leq \log^2 n$, where $\widetilde{O}(\cdot)$ hides $\mathsf{poly}(\log \log n)$ factors. These bounds improve upon the $O(\sqrt{k \log n})$ bound due to Banaszczyk [Random Struct. Algor., 1998]. For the Komlos problem, we give an $\widetilde{O}(\log^{1/4} n)$ bound, improving upon the previous $O(\sqrt{\log n})$ bound [Random Struct. Algor., 1998]. All of our results also admit efficient polynomial-time algorithms. To obtain these results, we consider a new notion of affine spectral-independence in designing random walks. In particular, our algorithms obtain the desired colorings via a discrete Brownian motion, guided by a semidefinite program (SDP). Besides standard constraints used in prior works, we add some extra affine spectral-independence constraints, which effectively decouple the evolution of discrepancy across different rows, and allow us to better control how many rows accumulate large discrepancy at any point during the process. This technique of ``decoupling via affine spectral-independence'' is quite general and may be of independent interest.

[12] arXiv:2508.03972 (交叉列表自 math-ph) [中文pdf, pdf, html, 其他]

标题： 费米子DGFF及其标度极限对数CFT 显示英文标题

标题： The fermionic DGFF and its scaling limit logCFT

David Adame-Carrillo, Wioletta M. Ruszel

主题： 数学物理 (math-ph) ; 概率 (math.PR)

在本文中，我们将费米子离散高斯自由场（fDGFF）的标度极限识别为二维的对数共形场理论（CFT）。我们首先建立了fDGFF局部可观测量的空间与辛费米子CFT局部场的空间之间的一一对应关系，这是一个中心电荷为$c = - 2$的对数CFT。这种对应关系是有意义的，因为当适当归一化时，fDGFF的相关函数在标度极限下收敛到相应的CFT相关函数。作为这些结果的应用，我们将均匀生成树和阿贝尔沙堆模型中的某些局部可观测量（及其标度极限）解释为辛费米子的局部场。 显示英文摘要

In this paper, we identify the scaling limit of the fermionic discrete Gaussian free field (fDGFF) as a logarithmic conformal field theory (CFT) in two dimensions. We first establish a one-to-one correspondence between the space of local observables of the fDGFF and the space of local fields of the symplectic fermions CFT, a logarithmic CFT with central charge $c = - 2$. This correspondence is meaningful in the sense that, when appropriately renormalised, the fDGFF correlation functions converge to corresponding CFT correlation functions in the scaling limit. As an application to these results, we interpret (the scaling limit of) certain local observables in the uniform spanning tree and the Abelian sandpile model as local fields of the symplectic fermions.

[13] arXiv:2508.04133 (交叉列表自 cs.CC) [中文pdf, pdf, 其他]

标题： 多项式时间采样尽管存在无序混沌 显示英文标题

标题： Polynomial-time sampling despite disorder chaos

Eric Ma, Tselil Schramm

主题： 计算复杂性 (cs.CC) ; 数据结构与算法 (cs.DS) ; 组合数学 (math.CO) ; 概率 (math.PR)

在抽样问题的实例上的一种分布被称为表现出传输无序混沌，如果对实例施加少量随机噪声会显著改变平稳分布（在Wasserstein距离下）。 为了提供一些抽样任务在平均情况下是困难的证据，最近的一系列工作表明，无序混沌足以排除“稳定”的抽样算法，例如梯度方法和某些扩散过程。 我们证明无序混沌并不排除在规范模型中的规范算法进行多项式时间抽样。 我们显示，在随机图$\boldsymbol{G} \sim G(n,1/2)$上高概率地满足以下条件：(1) 在 fugacity$\lambda = 1$下，$\boldsymbol{G}$上的hardcore模型表现出无序混沌，以及(2) Glauber动力学运行$O(n)$时间可以在 Wasserstein 距离下近似从$\boldsymbol{G}$上的 hardcore 模型中采样。 显示英文摘要

A distribution over instances of a sampling problem is said to exhibit transport disorder chaos if perturbing the instance by a small amount of random noise dramatically changes the stationary distribution (in Wasserstein distance). Seeking to provide evidence that some sampling tasks are hard on average, a recent line of work has demonstrated that disorder chaos is sufficient to rule out "stable" sampling algorithms, such as gradient methods and some diffusion processes. We demonstrate that disorder chaos does not preclude polynomial-time sampling by canonical algorithms in canonical models. We show that with high probability over a random graph $\boldsymbol{G} \sim G(n,1/2)$: (1) the hardcore model (at fugacity $\lambda = 1$) on $\boldsymbol{G}$ exhibits disorder chaos, and (2) Glauber dynamics run for $O(n)$ time can approximately sample from the hardcore model on $\boldsymbol{G}$ (in Wasserstein distance).

[14] arXiv:2508.04139 (交叉列表自 math-ph) [中文pdf, pdf, html, 其他]

标题： 实球面系综的实特征值分布渐进行为 显示英文标题

标题： Asymptotics of the real eigenvalue distribution for the real spherical ensemble

Peter J. Forrester

评论： 14页

主题： 数学物理 (math-ph) ; 概率 (math.PR)

实Ginibre球面系综由形式为$A B^{-1}$的随机矩阵组成，其中$A,B$是独立的标准实高斯$N \times N$矩阵。已知实特征值的期望数量为$\sqrt{N}$的量级。我们考虑在不同区域中存在$M$个实特征值的概率$p_{N.M}^{\rm r}$。 当$M$与$N$成比例时（大偏离），当$N$与$\sqrt{N}$成比例时（中间偏离），以及当$M$位于均值附近时（局部中心极限定理）。 在大偏离情况下，这是通过使用库仑气体形式实现的，并通过确定概率生成函数的主要渐近形式来实现中间偏离情况（局部中心极限区域的情况从早期工作中已知）。 此外，展示了中间偏离区域的左尾渐近行为与大偏离区域的右尾渐近行为之间的匹配，同时也展示了中间偏离区域的右尾与局部中心极限区域的概率主要形式之间的匹配。 我们还给出了$p_{N,0}^{\rm r}$的主要渐近形式，即没有实特征值的概率。 显示英文摘要

The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \times N$ matrices. The expected number of real eigenvalues is known to be of order $\sqrt{N}$. We consider the probability $p_{N.M}^{\rm r}$ that there are $M$ real eigenvalues in various regimes. These are when $M$ is proportional to $N$ (large deviations), when $N$ is proportional to $\sqrt{N}$ (intermediate deviations), and when $M$ is in the neighbourhood of the mean (local central limit theorem). This is done using a Coulomb gas formalism in the large deviations case, and by determining the leading asymptotic form of the generating function for the probabilities in the case of intermediate deviations (the local central limit regime was known from earlier work). Moreover a matching of the left tail asymptotics of the intermediate deviation regime with that of the right tail of the large deviation regime is exhibited, as is a matching of the right tail intermediate deviation regime with the leading order form of the probabilities in the local central limit regime. We also give the leading asymptotic form of $p_{N,0}^{\rm r}$, i.e. the probability of no real eigenvalues.

[15] arXiv:2508.04154 (交叉列表自 cond-mat.stat-mech) [中文pdf, pdf, 其他]

标题： 非平衡动力学与随机过程的首次通过性质：从布朗运动到活性粒子 显示英文标题

标题： Non-Equilibrium Dynamics and First-Passage Properties of Stochastic Processes: From Brownian Motion to Active Particles

Mathis Guéneau

评论： 博士论文于2025年6月16日在巴黎索邦大学答辩。252页

主题： 统计力学 (cond-mat.stat-mech) ; 无序系统与神经网络 (cond-mat.dis-nn) ; 软凝聚态物理 (cond-mat.soft) ; 数学物理 (math-ph) ; 概率 (math.PR)

在本论文中，我们开发了分析方法来研究由色噪声驱动的非平衡随机过程，即具有时间相关性的噪声。 与由白噪声驱动的过程（如布朗运动）相比，这些非马尔可夫过程带来了显著的分析挑战。 主要研究重点是活性粒子系统，特别是受到任意力作用的跑-停粒子。 我们通过后向福克-普朗克方程推导出其平均首次通过时间（MFPT）和从区间退出的概率的精确表达式。 值得注意的是，我们发现MFPT可以作为翻转率的函数进行优化。 此外，我们还研究了随机重置和切换扩散模型。 对于切换扩散模型——这是“布朗但非高斯扩散”的例子——我们使用更新方法和大偏差理论来推导各种可观测量的精确结果。 这些包括粒子位置的分布及其矩，还包括其累积量，累积量是表征非高斯涨落的关键可观测量。 值得注意的是，我们发现该模型与自由累积量之间存在意想不到的联系。 我们还通过使用Kesten变量研究了在谐波势存在的这些模型。 这种方法使我们能够写出稳态分布的积分方程，并在特定情况下求解。 此外，我们将塞格蒙德对偶性——这一概念在物理文献中并不广为人知——扩展到活性粒子、随机扩散模型、随机重置和连续时间随机游走。 这种对偶性建立了首次通过可观测量与对偶过程空间性质之间的直接关系，我们明确构造了这种对偶过程。 显示英文摘要

In this thesis, we develop analytical methods to study out-of-equilibrium stochastic processes driven by colored noise, i.e., noise with temporal correlations. These non-Markovian processes pose significant analytical challenges compared to processes driven by white noise, such as Brownian motion. A primary focus is on active particle systems, specifically the run-and-tumble particle subjected to an arbitrary force. We derive exact expressions for its mean first-passage time (MFPT) and exit probability from an interval using the backward Fokker-Planck equation. Remarkably, we find that the MFPT can be optimized as a function of the tumbling rate. Additionally, we investigate stochastic resetting and switching diffusion models. For switching diffusion models which are examples of "Brownian yet non-Gaussian diffusions", we use a renewal approach and large deviation theory to derive exact results for various observables. These include the distribution of the position of the particle and its moments, but also its cumulants which are key observables to characterize non-Gaussian fluctuations. Notably, we uncover an unexpected connection between this model and free cumulants. We also examine these models in the presence of a harmonic potential by using Kesten variables. This approach enables us to write an integral equation for the steady-state distribution, which we solve in specific cases. Furthermore, we extend Siegmund duality - a concept that is not widely known in the physics literature - to active particles, random diffusion models, stochastic resetting, and continuous-time random walks. This duality establishes a direct relation between first passage observables and the spatial properties of a dual process, which we explicitly construct.

[16] arXiv:2508.04608 (交叉列表自 cs.SI) [中文pdf, pdf, html, 其他]

标题： 几何和无标度网络中的 assortativity 显示英文标题

标题： Assortativity in geometric and scale-free networks

Marc Kaufmann, Ulysse Schaller, Thomas Bläsius, Johannes Lengler

主题： 社会与信息网络 (cs.SI) ; 概率 (math.PR)

网络的同配行为是指相似（或不相似）节点相互连接的倾向。 这种倾向可以影响网络的各种特性，例如其鲁棒性或传播过程的动力学特性。 在本文中，我们研究了现实世界网络以及基于潜在空间的具有重尾度分布的几种生成模型中的度同配性。 特别是，我们研究了Chung-Lu图和几何非均匀随机图（GIRGs）。 以往关于同配性的研究主要集中在使用皮尔逊同配系数来测量现实世界网络中的度同配性，尽管对此系数存在保留意见。 我们通过数学证明严格确认了这些保留意见，证明皮尔逊同配系数在具有足够重尾度分布的任何网络中都不测量同配性，而这是现实世界网络的典型特征。 此外，我们发现其他单值同配系数也不足以捕捉节点的布线偏好，因为这些偏好通常因节点度数而有很大差异。 因此，我们采用了一种更细致的方法，分析了连接节点的广泛条件和联合权重及度分布，既在现实世界网络中进行数值分析，也在生成图模型中进行数学分析。 我们提供了几种可视化结果的方法。 我们表明生成模型是同配性中立的，而许多现实世界网络并非如此。 因此，我们还提出了一种GIRG模型的扩展，该模型保留了由度分布和潜在空间引起的多种理想特性，但也表现出可调节的同配性。 我们对产生的模型进行了数学分析，并对其同配性进行了细致的量化。 显示英文摘要

The assortative behavior of a network is the tendency of similar (or dissimilar) nodes to connect to each other. This tendency can have an influence on various properties of the network, such as its robustness or the dynamics of spreading processes. In this paper, we study degree assortativity both in real-world networks and in several generative models for networks with heavy-tailed degree distribution based on latent spaces. In particular, we study Chung-Lu Graphs and Geometric Inhomogeneous Random Graphs (GIRGs). Previous research on assortativity has primarily focused on measuring the degree assortativity in real-world networks using the Pearson assortativity coefficient, despite reservations against this coefficient. We rigorously confirm these reservations by mathematically proving that the Pearson assortativity coefficient does not measure assortativity in any network with sufficiently heavy-tailed degree distributions, which is typical for real-world networks. Moreover, we find that other single-valued assortativity coefficients also do not sufficiently capture the wiring preferences of nodes, which often vary greatly by node degree. We therefore take a more fine-grained approach, analyzing a wide range of conditional and joint weight and degree distributions of connected nodes, both numerically in real-world networks and mathematically in the generative graph models. We provide several methods of visualizing the results. We show that the generative models are assortativity-neutral, while many real-world networks are not. Therefore, we also propose an extension of the GIRG model which retains the manifold desirable properties induced by the degree distribution and the latent space, but also exhibits tunable assortativity. We analyze the resulting model mathematically, and give a fine-grained quantification of its assortativity.

[17] arXiv:2508.04647 (交叉列表自 cond-mat.stat-mech) [中文pdf, pdf, html, 其他]

标题： 马尔可夫跳跃过程的路径可观测值的随机微积分：扩散和跳跃动力学的统一 显示英文标题

标题： Stochastic Calculus for Pathwise Observables of Markov-Jump Processes: Unification of Diffusion and Jump Dynamics

Lars Torbjørn Stutzer, Cai Dieball, Aljaž Godec

主题： 统计力学 (cond-mat.stat-mech) ; 概率 (math.PR)

路径可观测量——随机轨迹的泛函——是时间平均统计力学的核心，并且在热力学不等式如不确定性关系、速度限制和关联界限中起着关键作用。 它们在典型情况下提供了一种热力学推断的方法，当系统中所有耗散自由度无法实验上获取时。 到目前为止，专注于路径可观测量的理论发展主要集中在两个方向：扩散过程和马尔可夫跳跃动力学，几乎是以互不相关的方式进行的。 此外，即使对于扩散和跳跃动力学的相关结果，也是通过一系列不同的方法得出的，这些方法主要是间接的。 最近，随机微积分被证明可以为扩散过程的路径可观测量提供一种直接的方法，而对应的跳跃动力学框架却一直难以捉摸。 在我们的工作中，我们开发了一种与连续空间扩散完全平行的随机微积分，用于马尔可夫跳跃过程的路径可观测量。 我们为跳跃过程提出了一个“朗之万方程”，定义了广义的路径可观测量，并建立了它们的协变结构，从而全面考虑了瞬态和时间非齐次动力学。 我们以最一般的形式证明了已知的热力学不等式，并讨论了饱和条件。 我们确定了路径可观测量对一般（包括热）扰动的响应，并进行了连续极限，以实现扩散和跳跃动力学的完全统一。 我们的结果为离散状态下生成扩散模型的类比以及从波动轨迹中学习随机热力学开辟了新的途径。 显示英文摘要

Path-wise observables--functionals of stochastic trajectories--are at the heart of time-average statistical mechanics and are central to thermodynamic inequalities such as uncertainty relations, speed limits, and correlation-bounds. They provide a means of thermodynamic inference in the typical situation, when not all dissipative degrees of freedom in a system are experimentally accessible. So far, theories focusing on path-wise observables have been developing in two major directions, diffusion processes and Markov-jump dynamics, in a virtually disjoint manner. Moreover, even the respective results for diffusion and jump dynamics were derived with a patchwork of different approaches that are predominantly indirect. Stochastic calculus was recently shown to provide a direct approach to path-wise observables of diffusion processes, while a corresponding framework for jump dynamics remained elusive. In our work we develop, in an exact parallelism with continuous-space diffusion, a complete stochastic calculus for path-wise observables of Markov-jump processes. We formulate a "Langevin equation" for jump processes, define general path-wise observables, and establish their covariation structure, whereby we fully account for transients and time-inhomogeneous dynamics. We prove the known kinds of thermodynamic inequalities in their most general form and discus saturation conditions. We determine the response of path-wise observables to general (incl. thermal) perturbations and carry out the continuum limit to achieve the complete unification of diffusion and jump dynamics. Our results open new avenues in the direction of discrete-state analogs of generative diffusion models and the learning of stochastic thermodynamics from fluctuating trajectories.

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[18] arXiv:2206.01107 (替换) [中文pdf, pdf, html, 其他]

标题： 具有完全局部单调系数的随机偏微分方程的适定性 显示英文标题

标题： Well-posedness of stochastic partial differential equations with fully local monotone coefficients

Michael Röckner, Shijie Shang, Tusheng Zhang

评论： 此版本更新了我们已发表文章[Math. Ann., 2024, 390(3): 3419-3469]的早期预印本，包含了出版后发现的错误的更正。一份更正附在文档末尾。

期刊参考： 数学年鉴，2024年，390卷第3期：3419-3469

主题： 概率 (math.PR)

考虑带有完全局部单调系数的随机偏微分方程（SPDEs）在Gelfand三元组$V\subseteq H \subseteq V^*$中： \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], X(0) & = x\in H, \end{aligned} \right. \end{align*}其中 \begin{align*} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{align*}是可测映射，$L_2(U,H)$是从$U$到$H$的Hilbert-Schmidt算子空间，而$W$是一个$U$-圆柱形 维纳过程。 这样的SPDE包括应用领域如流体动力学等中的许多有趣模型。 在本文中，我们在完全局部单调性条件下建立了上述SPDE的适定性，解决了长期存在的开放问题。 扩散系数$B(t,\cdot)$的条件允许同时依赖于$H$-范数和$V$-范数。 在经典SPDE的情况下，这意味着$B(\cdot,\cdot)$也可以依赖于解的梯度。 适定性是通过伪单调性技术与紧致性论证的结合得到的。 显示英文摘要

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], X(0) & = x\in H, \end{aligned} \right. \end{align*} where \begin{align*} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{align*} are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process. Such SPDEs include many interesting models in applied fields like fluid dynamics etc. In this paper, we establish the well-posedness of the above SPDEs under fully local monotonicity condition solving a longstanding open problem. The conditions on the diffusion coefficient $B(t,\cdot)$ are allowed to depend on both the $H$-norm and $V$-norm. In the case of classical SPDEs, this means that $B(\cdot,\cdot)$ could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.

[19] arXiv:2212.13140 (替换) [中文pdf, pdf, html, 其他]

标题： 耗散测度值解到随机可压缩纳维-斯托克斯方程和无粘不可压缩极限 显示英文标题

标题： Dissipative Measure Valued Solutions to the Stochastic Compressible Navier-Stokes Equations and Inviscid-Incompressible Limit

Utsab Sarkar

评论： 期刊版本

主题： 概率 (math.PR) ; 偏微分方程分析 (math.AP)

我们引入了随机可压缩纳维-斯托克斯方程的耗散测度值鞅解的概念。 这些解从概率的角度来看是弱解，因为它们将驱动的维纳过程和概率空间作为解的整体部分。 然后，对于随机可压缩纳维-斯托克斯系统，我们建立了相对能量不等式，并由此证明了路径上弱强唯一性原理。 我们还利用相对能量不等式研究了原始方程组的无粘不可压缩极限。 显示英文摘要

We introduce a concept of dissipative measure valued martingale solutions for stochastic compressible Navier-Stokes equations. These solutions are weak from a probabilistic perspective, since they include both the driving Wiener process and the probability space as an integral part of the solution. Then, for the stochastic compressible Navier-Stokes system, we establish the relative energy inequality, and as a result, we demonstrate the path-wise weak-strong uniqueness principle. We also look at the inviscid-incompressible limit of the underlying system of equations using the relative energy inequality.

[20] arXiv:2307.11372 (替换) [中文pdf, pdf, html, 其他]

标题： 论玻尔兹曼分布的起源 显示英文标题

标题： On the origin of the Boltzmann distribution

Fedor Sandomirskiy, Omer Tamuz

主题： 概率 (math.PR) ; 统计力学 (cond-mat.stat-mech) ; 群论 (math.GR)

玻尔兹曼分布族在统计力学中用于描述给定温度下系统的状态分布。 我们给出了该族的一个新特征，即作为满足非耦合系统独立性的唯一一族。 该定理简化为关于自然数上有限支撑概率测度的卷积半群的自同态的陈述，或者，等价地，关于具有非负系数的多项式的乘法半群的自同态的陈述。 显示英文摘要

The family of Boltzmann distributions is used in statistical mechanics to describe the distribution of states in systems with a given temperature. We give a novel characterization of this family as the unique one satisfying independence for uncoupled systems. The theorem boils down to a statement about endomorphisms of the convolution semi-group of finitely supported probability measures on the natural numbers, or, alternatively, about endomorphisms of the multiplicative semi-group of polynomials with non-negative coefficients.

[21] arXiv:2401.09338 (替换) [中文pdf, pdf, html, 其他]

标题： 关于时间非齐次跳跃驱动SDE的$\varepsilon$-Euler-Maruyama格式 显示英文标题

标题： On the $\varepsilon$-Euler-Maruyama scheme for time-inhomogeneous jump-driven SDEs

Mireille Bossy, Paul Maurer

主题： 概率 (math.PR)

我们考虑由时间非齐次泊松随机测度驱动的具有跳跃积分项的一类广义SDE。 我们为此类SDE提出一个双参数的欧拉型方案，并证明在关于$L^p(\Omega)$-范数的强收敛和弱收敛方面具有最优速率，考虑在$n$均匀时间步长上的积分。 在此背景下需要解决的主要问题之一是当噪声结构不能再表示为随机变量的增量时的近似。 我们将Asmussen-Rosiński方法扩展到跳跃系数完全依赖和时间依赖的泊松补偿的情况，对于小于$\varepsilon$的跳跃贡献使用适当的高斯替代，而对于大跳跃贡献进行精确模拟。 对于任何$p \geq 2$，在控制过程的$L^p$-矩所需的假设下，我们得到一个阶为$1/p$的强收敛速率。 在系数的标准正则性假设下，我们得到一个弱收敛速度为$1/n+\varepsilon^{3-\beta}$，其中$\beta$是底层Lévy测度的Blumenthal-Getoor指数。 我们将该方案与Rubenthaler的方法进行比较，在该方法中忽略小于$\varepsilon$的跳跃，同时也提供了该情况下的强收敛和弱收敛速度。 随后通过数值实验验证了这些理论收敛速度。 我们将该模型应用于与湍流中刚性纤维动力学相关的异常扩散模型。 显示英文摘要

We consider a class of general SDEs with a jump integral term driven by a time-inhomogeneous Poisson random measure. We propose a two-parameters Euler-type scheme for this SDE class and prove an optimal rate for the strong convergence with respect to the $L^p(\Omega)$-norm and for the weak convergence, considering integration over $n$ uniform time-steps. One of the primary issues to address in this context is the approximation of the noise structure when it can no longer be expressed as the increment of random variables. We extend the Asmussen-Rosi\'nski approach to the case of a fully dependent jump coefficient and time-dependent Poisson compensation, handling contribution of jumps smaller than $\varepsilon$ with an appropriate Gaussian substitute and exact simulation for the large jumps contribution. For any $p \geq 2$, under hypotheses required to control the $L^p$-moments of the process, we obtain a strong convergence rate of order $1/p$. Under standard regularity hypotheses on the coefficients, we obtain a weak convergence rate of $1/n+\varepsilon^{3-\beta}$, where $\beta$ is the Blumenthal-Getoor index of the underlying L\'evy measure. We compare this scheme with the Rubenthaler's approach where the jumps smaller than $\varepsilon$ are neglected, providing strong and weak rates of convergence in that case too. The theoretical rates are confirmed by numerical experiments afterwards. We apply this model class for some anomalous diffusion model related to the dynamics of rigid fibres in turbulence.

[22] arXiv:2412.04539 (替换) [中文pdf, pdf, html, 其他]

标题： 计数最小割集和$p_c<1$ 显示英文标题

标题： Counting minimal cutsets and $p_c<1$

Philip Easo, Franco Severo, Vincent Tassion

评论： 13页。已接受发表于《数学论坛》Pi版

主题： 概率 (math.PR) ; 数学物理 (math-ph) ; 组合数学 (math.CO) ; 群论 (math.GR)

我们证明了关于一般图上渗流的两个结果。 - 我们建立了经典佩尔斯论证的逆定理：如果（均匀）渗流的临界参数满足$p_c<1$，则将给定顶点与无限远分隔的大小为$n$的最小割集的数量在$n$上呈指数级上限。 这解决了 Babson 和 Benjamini 于 1999 年提出的猜想。 - 我们证明了对于每个均匀暂态图，$p_c<1$。 这解决了 Duminil-Copin、Goswami、Raoufi、Severo 和 Yadin 提出的问题，并提供了另一种证明方式，即对于每个超线性增长的传递图，$p_c<1$。 显示英文摘要

We prove two results concerning percolation on general graphs. - We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies $p_c<1$, then the number of minimal cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This resolves a conjecture of Babson and Benjamini from 1999. - We prove that $p_c<1$ for every uniformly transient graph. This solves a problem raised by Duminil-Copin, Goswami, Raoufi, Severo and Yadin, and provides a new proof that $p_c<1$ for every transitive graph of superlinear growth.

[23] arXiv:2502.02579 (替换) [中文pdf, pdf, html, 其他]

标题： 线性上激活随机游走的超可加性和密度猜想的新证明 显示英文标题

标题： A new proof of superadditivity and of the density conjecture for Activated Random Walks on the line

Nicolas Forien

评论： 18页。经过小幅修改的改进展示，增加了一张图表

主题： 概率 (math.PR)

在两项最近的研究中，Hoffman、Johnson 和 Junge 证明了在一维激活随机游走模型中密度猜想、曲棍球棒猜想和球体猜想，展示了该模型中几种不同临界密度定义之间的等价性。这建立了一种自组织临界性，最初是针对阿贝尔沙堆模型预测的。他们的证明使用了与具有超加性的渗流过程的比较。我们在此提供这些猜想的不同证明，基于我们直接为激活随机游走建立的新超加性性质，而不依赖于渗流过程。这种更基础的方法得到的界限不如 Hoffman、Johnson 和 Junge 开发的渗流技术精确，但它可能为超越一维设置开辟新的前景。 显示英文摘要

In two recent works, Hoffman, Johnson and Junge proved the density conjecture, the hockey stick conjecture and the ball conjecture for Activated Random Walks in dimension one, showing an equality between several different definitions of the critical density of the model. This establishes a kind of self-organized criticality, which was originally predicted for the Abelian Sandpile Model. Their proof uses a comparison with a percolation process, which exhibits superadditivity. We present here a different proof of these conjectures, based on a new superadditivity property that we establish directly for Activated Random Walks, without relying on a percolation process. This more elementary approach yields less precise bounds than the percolation technology developed by Hoffman, Johnson and Junge, but it might open new perspectives to go beyond the one-dimensional setting.

[24] arXiv:2503.21737 (替换) [中文pdf, pdf, html, 其他]

标题： 高密度流形上的Voronoi渗流 显示英文标题

标题： High-intensity Voronoi percolation on manifolds

Tillmann Bühler, Barbara Dembin, Ritvik Ramanan Radhakrishnan, Franco Severo

评论： 42页，3张图。由于错误移除了唯一性表征

主题： 概率 (math.PR)

我们研究大类的$d$-维黎曼流形上的Voronoi渗流，该类包括双曲空间$\mathbb{H}^d$，$d\geq 2$。 我们证明当底层泊松点过程的强度$\lambda$趋向于无穷大时，两个临界参数$p_c(M,\lambda)$和$p_u(M,\lambda)$都收敛到欧几里得临界参数$p_c(\mathbb{R}^d)$。 这将Hansen & Müller在特殊情形$M=\mathbb{H}^2$下的近期结果扩展到任意维数的流形的一般类中。 我们证明中的一个关键步骤，可能具有独立兴趣，是证明如果$M$是单连通且单端的，那么由流形$M$上的一般铺砌类所诱导的嵌入图具有连通的最小割集。 特别是，这个结果适用于$\varepsilon$-网，使我们能够实施一个“细粒化”论证。 显示英文摘要

We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $\lambda$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,\lambda)$ and $p_u(M,\lambda)$ converge to the Euclidean critical parameter $p_c(\mathbb{R}^d)$. This extends a recent result of Hansen & M\"uller in the special case $M=\mathbb{H}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\varepsilon$-nets, allowing us to implement a "fine-graining" argument.

[25] arXiv:2503.21768 (替换) [中文pdf, pdf, html, 其他]

标题： 通过 germ order 对分支随机游走和谣言过程的结果 显示英文标题

标题： Results on branching random walks and rumor processes via germ order

Daniela Bertacchi, Fabio Zucca

评论： 28页

主题： 概率 (math.PR)

germ顺序是一种通过比较过程生成函数定义的非标准随机顺序。 这种顺序最初是为具有恒定繁殖律和独立后代扩散的分支随机游走引入的，这些过程由一维生成函数表征。 在本工作中，我们研究了该概念扩展到由多维生成函数表征的过程的性质，例如一般的分支随机游走和谣言过程。 特别是，我们使用germ顺序来表征某些具有非均匀繁殖/传播律的分支随机游走和谣言过程的行为。 显示英文摘要

Germ order is a non-standard stochastic order defined through the comparison of the generating functions of the processes. This order was first introduced for branching random walks with a constant breeding law and independent dispersal of offspring, which are characterized by a one-dimensional generating function. In this work, we investigate the properties of the extension of this concept to processes characterized by a multidimensional generating function, such as general branching random walks and rumor processes. In particular, we use germ ordering to characterize the behavior of certain branching random walks and rumor processes with inhomogeneous breeding/transmitting laws.

[26] arXiv:2505.11457 (替换) [中文pdf, pdf, html, 其他]

标题： 高温度伊辛模型交叉的噪声敏感性 显示英文标题

标题： Noise sensitivity of crossings for high temperature Ising model

Vincent Tassion, Hugo Vanneuville

评论： 44页，进行了小幅度修改

主题： 概率 (math.PR) ; 数学物理 (math-ph)

考虑在三角形格子上的伊辛模型中，存在从左到右的$+$跨越的事件。 我们在亚临界区域$\beta<\beta_c$下的Glauber动力学$t \mapsto \sigma_t$中证明了该事件是噪声敏感的。 我们依赖于之前工作[TV23]中的非谱方法。 在这个更一般的设置中，一个重要方面是研究对$(\sigma_0,\sigma_t)$的研究，特别是建立诸如有限能量和空间混合等性质。 显示英文摘要

Consider the event that there is a $+$ crossing from left to right in a box for the Ising model on the triangular lattice. We show that this event is noise sensitive under Glauber dynamics $t \mapsto \sigma_t$ in the subcritical regime $\beta<\beta_c$. We rely on the non-spectral approach from our previous work [TV23]. An important aspect in this more general setup is the study of the pair $(\sigma_0,\sigma_t)$ and in particular the establishment of properties such as finite-energy and spatial mixing.

[27] arXiv:2507.02676 (替换) [中文pdf, pdf, html, 其他]

标题： 关于临界高维渗透中的环 显示英文标题

标题： On Loops in critical high-dimensional percolation

Amelia Carpenter, Wendelin Werner

评论： 20页，V2：小幅度修改

主题： 概率 (math.PR)

我们讨论高维临界伯努利渗流的以下类型的结果：在大方框中包含大（无自交）环的簇的集合是紧的。 这些大环的集合具有可以与布朗环浴相关联的标度极限（每个这种非典型的含环簇在某种意义上只包含一个大的无自交环，即任何两个这样的环都会非常接近）。 这一特性与已知的“典型”渗流簇的大量增殖形成对比（即，在一个给定的方框中的许多大簇中，只有少数会包含大小与方框相当的环）。 显示英文摘要

We discuss the following type of results about critical Bernoulli percolation in high dimensions: The collection of clusters that do contain large (self-avoiding) loops in a large box is tight. The collection of these large loops has scaling limits that one will be able to relate to Brownian loop-soups (each of these atypical loop-containing clusters does in some sense only contain one large self-avoiding loop in the sense that any two such loops will be very close). This feature contrasts with the known proliferation of "typical" percolation clusters (i.e., among the many large clusters in a given box, only a handful will contain a loop of size comparable to the box).

[28] arXiv:2305.14496 (替换) [中文pdf, pdf, html, 其他]

标题： 通过中偏差理论进行最优学习 显示英文标题

标题： Optimal Learning via Moderate Deviations Theory

Arnab Ganguly, Tobias Sutter

评论： 35页，3图

主题： 机器学习 (stat.ML) ; 优化与控制 (math.OC) ; 概率 (math.PR) ; 统计理论 (math.ST)

本文提出了一种统计上最优的方法，用于在广泛的模型中使用置信区间学习函数值，包括一般非参数估计的期望损失，该损失被描述为随机规划问题或各种SDE模型。 更准确地说，我们通过使用基于中偏差原理的方法，系统地构建了高度精确的置信区间。 结果表明，所提出的置信区间在统计上是最优的，因为它们满足关于指数精度、最小性、一致性、误表征概率和最终一致最准确（UMA）性质的标准。 这种方法建议的置信区间被表述为鲁棒优化问题的解，其中不确定性通过由数据生成过程引起的中偏差率函数来表达。 我们证明了对于许多模型，这些优化问题即使在无限维的情况下，也可以转化为可处理的有限凸规划问题。 显示英文摘要

This paper proposes a statistically optimal approach for learning a function value using a confidence interval in a wide range of models, including general non-parametric estimation of an expected loss described as a stochastic programming problem or various SDE models. More precisely, we develop a systematic construction of highly accurate confidence intervals by using a moderate deviation principle-based approach. It is shown that the proposed confidence intervals are statistically optimal in the sense that they satisfy criteria regarding exponential accuracy, minimality, consistency, mischaracterization probability, and eventual uniformly most accurate (UMA) property. The confidence intervals suggested by this approach are expressed as solutions to robust optimization problems, where the uncertainty is expressed via the underlying moderate deviation rate function induced by the data-generating process. We demonstrate that for many models these optimization problems admit tractable reformulations as finite convex programs even when they are infinite-dimensional.

[29] arXiv:2506.11794 (替换) [中文pdf, pdf, 其他]

标题： ALEA IACTA EST：一种用于可手动执行的随机实验的声明式领域特定语言 显示英文标题

标题： ALEA IACTA EST: A Declarative Domain-Specific Language for Manually Performable Random Experiments

Baltasar Trancón y Widemann (Brandenburg University of Applied Sciences), Markus Lepper (semantics gGmbH)

评论： 在《TFPiE 2025论文集》中，arXiv:2508.02305

期刊参考： EPTCS 424，2025，第67-86页

主题： 编程语言 (cs.PL) ; 概率 (math.PR)

随机实验如果简单明了，足以由人类代理执行，在初等概率论的教学以及游戏中占据重要地位。 我们提出了Alea，一种用于指定随机实验的领域特定语言。 Alea代码可以进行静态分析以获取和检查结果的概率分布，或者使用源伪随机性进行执行以进行模拟或作为游戏助手。 该语言旨在便于非专业程序员使用，特别是初等概率论的学生以及机会游戏的玩家和设计者，其重点在于函数式编程和基础数学中的常见概念。 语言的设计和运行时环境的实现仍在进行中。 显示英文摘要

Random experiments that are simple and clear enough to be performed by human agents feature prominently in the teaching of elementary stochastics as well as in games. We present Alea, a domain-specific language for the specification of random experiments. Alea code can either be analyzed statically to obtain and inspect probability distributions of outcomes, or be executed with a source pseudo-randomness for simulation or as a game assistant. The language is intended for ease of use by non-expert programmers, in particular students of elementary stochastics, and players and designers of games of chance, by focusing on concepts common to functional programming and basic mathematics. Both the design of the language and the implementation of runtime environments are work in progress.

[30] arXiv:2506.22883 (替换) [中文pdf, pdf, html, 其他]

标题： 关于无共轭点流形在无穷远处的狄利克雷问题和泊松边界 显示英文标题

标题： On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points

Fei Liu, Yinghan Zhang

评论： 50页，6个图。欢迎提出意见

主题： 微分几何 (math.DG) ; 概率 (math.PR)

在本文中，我们研究了在没有共轭点的单连通黎曼流形$\widetilde{M}$上有界调和函数的存在性问题，该流形可以通过理想边界$\widetilde{M}(\infty)$进行紧化。 设$\widetilde{M}$是一个满足公理$2$的均匀可见流形，或是一个无聚焦点的秩$1$流形，假设$\Gamma$是$Iso(\widetilde{M})$的共紧离散子群，我们证明对于$\widetilde{M}(\infty)$上的给定连续函数，存在一个到$\widetilde{M}$的调和延拓。 并且此外，当$\widetilde{M}$是一个没有聚焦点的秩$1$流形时，布朗运动在$\widetilde{M}(\infty)$上定义了一族调和测度$\nu_{\ast}$，我们证明$(\widetilde{M}(\infty),\nu_{\ast})$与$\Gamma$的泊松边界是同构的。 显示英文摘要

In this paper, we investigate the problem of the existence of the bounded harmonic functions on a simply connected Riemannian manifold $\widetilde{M}$ without conjugate points, which can be compactified via the ideal boundary $\widetilde{M}(\infty)$. Let $\widetilde{M}$ be a uniform visibility manifold which satisfy the Axiom $2$, or a rank $1$ manifold without focal points, suppose that $\Gamma$ is a cocompact discrete subgroup of $Iso(\widetilde{M})$, we show that for a given continuous function on $\widetilde{M}(\infty)$, there exists a harmonic extension to $\widetilde{M}$. And furthermore, when $\widetilde{M}$ is a rank $1$ manifold without focal points, the Brownian motion defines a family of harmonic measures $\nu_{\ast}$ on $\widetilde{M}(\infty)$, we show that $(\widetilde{M}(\infty),\nu_{\ast})$ is isomorphic to the Poisson boundary of $\Gamma$.

[31] arXiv:2508.02707 (替换) [中文pdf, pdf, html, 其他]

标题： 传输噪声在$\mathbb{S}^2$上的扩散行为 显示英文标题

标题： Diffusive behavior of transport noise on $\mathbb{S}^2$

Sagy Ephrati, Erik Jansson, Andrea Papini

评论： 13页，2张图。欢迎所有评论！

主题： 数值分析 (math.NA) ; 概率 (math.PR) ; 流体动力学 (physics.flu-dyn)

我们从理论上和数值上研究球面上由输运噪声引起的扩散。 在环面之前的分析表明，适当选择的欧拉方程中的输运噪声会导致类似于纳维-斯托克斯方程的扩散行为。 在这里，我们分析球面上由噪声引起的微分椭圆算子耗散的动力学，并表征其能量和涡旋度衰减特性。 通过使用Zeitlin离散化的结构保持数值模拟，我们证明适当缩放的输运噪声会引发能量耗散，同时保持涡旋度和共伴随轨道。 所提出的分析为输运噪声的进一步理论研究奠定了基础，并支持将输运噪声模型校准为地理流体模拟中未解析过程的参数化方法。 显示英文摘要

We investigate theoretically and numerically transport noise-induced diffusion in flows on the sphere. Previous analysis on the torus demonstrated that suitably chosen transport noise in the Euler equations leads to diffusive behavior resembling the Navier--Stokes equations. Here, we analyze dynamics on the sphere with noise-induced differential elliptic operator dissipation and characterize their energy and enstrophy decay properties. Through structure-preserving numerical simulations with the Zeitlin discretization, we demonstrate that appropriately scaled transport noise induces energy dissipation while preserving enstrophy and coadjoint orbits. The presented analysis lays a groundwork for further theoretical investigation of transport noise and supports the calibration of transport noise models as a parametrization for unresolved processes in geophysical fluid simulations.