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- [1] arXiv:2507.13512 [中文pdf, pdf, html, 其他]
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标题： Hadamard分数布朗运动：路径性质和维纳积分标题： Hadamard fractional Brownian motion: path properties and Wiener integration评论： 36页，2图主题： 概率 (math.PR)
所谓的Hadamard分数阶布朗运动，如Beghin等人（2025）通过Hadamard分数阶算子定义的，是一个高斯过程，它与标准布朗运动（如一维分布）有一些相似的性质。然而，它在许多其他特征上也类似于分数阶布朗运动，例如自相似性、长/短记忆特性、维纳积分表示。Hadamard分数阶布朗运动中的对数核代表了该过程的一个非常具体且有趣的特点。我们在此分析该过程轨迹的一些性质（即霍尔德连续性、准螺旋行为、幂次变分、局部非确定性），这些性质本身就很有趣，并且是针对该过程进行维纳积分的基础。相应的积分已经得到了很好的发展，逆表示也被构建出来。我们将推导出的“乘法Sonine对”应用于Hadamard分数阶布朗运动的再生核希尔伯特空间的处理，并由此建立了一个迭代对数定律。
The so-called Hadamard fractional Brownian motion, as defined in Beghin et al. (2025) by means of Hadamard fractional operators, is a Gaussian process which shares some properties with standard Brownian motion (such as the one-dimensional distribution). However, it also resembles the fractional Brownian motion in many other features as, for instance, self-similarity, long/short memory property, Wiener-integral representation. The logarithmic kernel in the Hadamard fractional Brownian motion represents a very specific and interesting aspect of this process. Our aim here is to analyze some properties of the process' trajectories (i.e. H\"{o}lder continuity, quasi-helix behavior, power variation, local nondeterminism) that are both interesting on their own and serve as a basis for the Wiener integration with respect to it. The respective integration is quite well developed, and the inverse representation is also constructed. We apply the derived ``multiplicative Sonine pairs'' to the treatment of the Reproducing Kernel Hilbert Space of the Hadamard fractional Brownian motion, and, as a result, we establish a law of iterated logarithm.
- [2] arXiv:2507.13747 [中文pdf, pdf, html, 其他]
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标题： 马尔可夫微积分和随机微分方程标题： Malliavin Calculus and Stochastic Differential Equations主题： 概率 (math.PR)
本文致力于研究具有有界博雷尔漂移项 b 的随机微分方程。 我们首先指出，P. 马里亚纳原始的分部积分公式可用于处理关于空间变量的导数，然后我们得到了热核乘积与马里亚纳微积分中迭代散度之间的联系。 通过 b 的 L-无穷范数，获得了随机微分方程解的导数的显式估计；作为结果，我们证明了该随机微分方程在索伯列夫空间中定义了一个连续的映射流。
This paper is devoted to a study on SDEs with a bounded Borel drift b. We first remark that the original integration by parts formula due to P. Malliavin can be used to deal with derivatives with respect to space variables, then we obtain a link between the product of heat kernels and iterated divergences in Malliavin calculus. An explicit estimate for the derivative of solutions to SDE is obtained in terms of the L-infinity norm of b; as a result, we prove that the SDE defines a continuous flow of maps in Sobolev spaces.
- [3] arXiv:2507.13799 [中文pdf, pdf, 其他]
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标题： 调制泊松-狄利克雷扩散，源于缓慢相的包含过程标题： Modulated Poisson-Dirichlet diffusions arising from inclusion processes with a slow phase评论： 44页。欢迎提出意见！主题： 概率 (math.PR) ; 统计力学 (cond-mat.stat-mech)
我们研究带有额外慢相的平均场包含过程，在该慢相中，粒子相互作用以与系统大小倒数成比例的消失速率发生。 在热力学极限下，这些系统在高粒子密度下表现出凝聚，形成尺寸发散的簇。 我们的主要结果提供了包含过程对一种新型两组分无限维随机扩散的法律收敛，描述了固态凝聚相和微观流体相的共同演化。 特别是，我们建立了两个相之间的非平凡质量交换。 所得的标度极限扩展了泊松-狄利克雷扩散（Ethier 和 Kurtz，1981），引入了一个额外的控制过程来调节其参数。 我们的结果基于生成器差异的经典估计，在这种情况下产生非消失的确定性误差界限。 我们通过显示瞬时凝聚来提供缺失的概率要素，其中粒子簇立即集中在消失的体积分数上。 我们进一步在紧致状态空间上一般地建立了极限动力学的适定性，作为 Feller 过程。
We study mean-field inclusion processes with an additional slow phase, in which particle interactions occur at a vanishing rate proportional to the inverse system size. In the thermodynamic limit, such systems exhibit condensation at high particle density, forming clusters of diverging size. Our main result provides convergence in law of inclusion processes to a novel two-component infinite-dimensional stochastic diffusion, describing the co-evolution of the solid condensed and microscopic fluid phase. In particular, we establish non-trivial mass exchange between the two phases. The resulting scaling limit extends the Poisson-Dirichlet diffusion (Ethier and Kurtz, 1981), introducing an additional control process that modulates its parameters. Our result builds on classical estimates of generator differences, which in this setting yield non-vanishing deterministic error bounds. We provide the missing probabilistic ingredient by showing instantaneous condensation, with particle clusters concentrating on a vanishing volume fraction immediately. We further establish the well-posedness of the limiting dynamics generally as Feller processes on a compact state space.
- [4] arXiv:2507.13922 [中文pdf, pdf, 其他]
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标题： 乘法布朗运动在一般线性群上的强收敛性标题： Strong Convergence of Multiplicative Brownian Motions on the General Linear Group评论： 44页主题： 概率 (math.PR) ; 数学物理 (math-ph)
我们考虑由Driver-Hall-Kemp引入的广义线性群上的乘法布朗运动族$G_{\lambda,\tau}$。 它们由底层椭圆布朗运动的实方差$\lambda\in \mathbb{R}$和复协方差$\tau \in \mathbb{C}$参数化。 我们证明了$G_{\lambda,\tau}$的有限维边缘几乎必然强收敛于Hall-Ho引入的相应自由乘法布朗运动：当维度趋于无穷时，不仅非交换分布几乎必然收敛，算子范数也如此。 这个结果推广了Collins-Dahlqvist-Kemp针对特殊情形$(\lambda,\tau)=(1,0)$的工作，该情形对应于酉群上的布朗运动。 实际上，当考虑乘法布朗运动族$G_{\lambda,\tau}$与一组几乎必然收敛的确定性矩阵一起时，这种强收敛仍然成立。
We consider the family of multiplicative Brownian motions $G_{\lambda,\tau}$ on the general linear group introduced by Driver-Hall-Kemp. They are parametrized by the real variance $\lambda\in \mathbb{R}$ and the complex covariance $\tau \in \mathbb{C}$ of the underlying elliptic Brownian motion. We show the almost sure strong convergence of the finite-dimensional marginals of $G_{\lambda,\tau}$ to the corresponding free multiplicative Brownian motion introduced by Hall-Ho: as the dimension tends to infinity, not only does the noncommutative distribution converge almost surely, but the operator norm does as well. This result generalizes the work of Collins-Dahlqvist-Kemp for the special case $(\lambda,\tau)=(1,0)$ which corresponds to the Brownian motion on the unitary group. Actually, this strong convergence remains valid when the family of multiplicative Brownian motions $G_{\lambda,\tau}$ is considered alongside a family of strongly converging deterministic matrices.
- [5] arXiv:2507.13990 [中文pdf, pdf, html, 其他]
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标题： 一个Durrett-Remenik粒子系统在$\mathbb{R}^d$标题： A Durrett-Remenik particle system in $\mathbb{R}^d$主题： 概率 (math.PR)
本文研究了在$\mathbb{R}^d$中的静止粒子分支-选择模型，该模型由 Durrett 和 Remenik 在维度$1$中引入。对适应度函数$F$和非均匀分支分布的假设是温和的。宏观密度的演化方程由一个在$\mathbb{R}^d$中的积分微分自由边界问题给出，其中自由边界表示种群中的最小$F$-值。主要结果是根据这个自由边界问题的唯一解来表征经验测度过程的概率极限。
This paper studies a branching-selection model of motionless particles in $\mathbb{R}^d$, with nonlocal branching, introduced by Durrett and Remenik in dimension $1$. The assumptions on the fitness function, $F$, and on the inhomogeneous branching distribution, are mild. The evolution equation for the macroscopic density is given by an integro-differential free boundary problem in $\mathbb{R}^d$, in which the free boundary represents the least $F$-value in the population. The main result is the characterization of the limit in probability of the empirical measure process in terms of the unique solution to this free boundary problem.
- [6] arXiv:2507.14008 [中文pdf, pdf, html, 其他]
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标题： 一维气体在边缘的大偏差以及高温下三对角随机矩阵标题： Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature评论： 欢迎评论！主题： 概率 (math.PR) ; 数学物理 (math-ph)
我们考虑一个模型，其中包含受双体排斥相互作用的 $N$ 个被限制粒子，即一维对数或 Riesz 气体。 我们感兴趣的是所谓的 \textit{高温} 范围， \textit{即}其中逆温度 $\beta_N$随 $N\beta_N\rightarrow2P>0$变化。 在对数情况下，我们建立了当最大粒子 $x_\mathrm{max}$适当缩放时的大偏差（LD）原理和中等偏差估计。 我们的结果是对[Ben-Arous, Dembo, Guionnet 2001]和[Pakzad 2020]中结果的扩展，其中在固定$\beta_N=\beta>0$和$\beta_N\gg N^{-1}$的情况下，分别对$\beta$-ensemble 中的最大粒子进行了此类估计。我们证明了相应的速率函数与独立同分布粒子的情况相同。我们还提供了瑞利情形下的LD估计。此外，我们考虑了具有独立条目且尾部为高斯的对称三对角随机矩阵的相关模型；对于这些模型，我们建立了顶部特征值的LD原理。在条目的一种特定情况下，我们恢复了对数气体最大粒子的结果。我们证明了LD是由少数条目取异常大值所引起的。
We consider a model for a gas of $N$ confined particles subject to a two-body repulsive interaction, namely the one-dimensional log or Riesz gas. We are interested in the so-called \textit{high temperature} regime, \textit{ie} where the inverse temperature $\beta_N$ scales as $N\beta_N\rightarrow2P>0$. We establish, in the log case, a large deviation (LD) principle and moderate deviations estimates for the largest particle $x_\mathrm{max}$ when appropriately rescaled . Our result is an extension of [Ben-Arous, Dembo, Guionnet 2001] and [Pakzad 2020 where such estimates were shown for the largest particle of the $\beta$-ensemble respectively at fixed $\beta_N=\beta>0$ and $\beta_N\gg N^{-1}$. We show that the corresponding rate function is the same as in the case of iid particles. We also provide LD estimates in the Riesz case. Additionally, we consider related models of symmetric tridiagonal random matrices with independent entries having Gaussian tails; for which we establish the LD principle for the top eigenvalue. In a certain specialization of the entries, we recover the result for the largest particle of the log-gas. We show that LD are created by a few entries taking abnormally large values.
新提交 (展示 6 之 6 条目 )
- [7] arXiv:2507.13503 (交叉列表自 cond-mat.stat-mech) [中文pdf, pdf, 其他]
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标题： Potts格点规范理论的广义聚类算法标题： Generalized cluster algorithms for Potts lattice gauge theory主题： 统计力学 (cond-mat.stat-mech) ; 计算几何 (cs.CG) ; 数学物理 (math-ph) ; 概率 (math.PR)
蒙特卡罗算法，如Swendsen-Wang和入侵簇算法，在采样伊辛模型和庞茨模型时，比单自旋Glauber动力学的渐近速度更快。 在这里，我们通过一种称为平面元随机簇模型的$2$维单元表示方法，将这两种算法推广以采样庞茨格点规范理论。 入侵簇算法通过实现基于同调渗透的停止条件来针对临界状态下的庞茨格点规范理论，即在环面上出现贯穿表面。 在立方$4$维环面的$\mathbb Z_2$和$\mathbb Z_3$格点规范理论的模拟表明，这两种推广算法的自相关衰减速度比单自旋动力学快得多，并且允许在至少$40$线性尺度的$4$维环面上进行高效采样。
Monte Carlo algorithms, like the Swendsen-Wang and invaded-cluster, sample the Ising and Potts models asymptotically faster than single-spin Glauber dynamics do. Here, we generalize both algorithms to sample Potts lattice gauge theory by way of a $2$-dimensional cellular representation called the plaquette random-cluster model. The invaded-cluster algorithm targets Potts lattice gauge theory at criticality by implementing a stopping condition defined in terms of homological percolation, the emergence of spanning surfaces on the torus. Simulations for $\mathbb Z_2$ and $\mathbb Z_3$ lattice gauge theories on the cubical $4$-dimensional torus indicate that both generalized algorithms exhibit much faster autocorrelation decay than single-spin dynamics and allow for efficient sampling on $4$-dimensional tori of linear scale at least $40$.
- [8] arXiv:2507.13763 (交叉列表自 q-fin.RM) [中文pdf, pdf, html, 其他]
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标题： 引出与法律不变泛函相关的参考度量标题： Eliciting reference measures of law-invariant functionals主题： 风险管理 (q-fin.RM) ; 概率 (math.PR)
规律不变泛函是风险管理的核心，它在无原子参考概率测度下，为具有相同分布的随机前景赋予相同的值。该测度通常被假设为固定不变的。在这里，我们采用相反的视角：仅给定观测到的泛函值，我们的目标是要么恢复参考测度，要么在不先验满足该性质时，确定一个候选测度来测试规律不变性。我们的方法基于对在规律不变域上定义的规律不变泛函的一个关键观察。这些泛函在有符号测度的对偶空间中定义了下（上）支撑集，并且这些支撑集的上确界（如果存在）是参考测度的标量倍数。在特定情况下，这一观察可以表述为三明治定理。我们通过对一些著名案例的详细分析来说明该方法：熵风险测度、预期损失和风险价值。对于后者，我们的引出程序最初由于支持集极值的简单性而失败。因此，我们开发了一个合适的修改方案。
Law-invariant functionals are central to risk management and assign identical values to random prospects sharing the same distribution under an atomless reference probability measure. This measure is typically assumed fixed. Here, we adopt the reverse perspective: given only observed functional values, we aim to either recover the reference measure or identify a candidate measure to test for law invariance when that property is not {\em a priori} satisfied. Our approach is based on a key observation about law-invariant functionals defined on law-invariant domains. These functionals define lower (upper) supporting sets in dual spaces of signed measures, and the suprema (infima) of these supporting sets -- if existent -- are scalar multiples of the reference measure. In specific cases, this observation can be formulated as a sandwich theorem. We illustrate the methodology through a detailed analysis of prominent examples: the entropic risk measure, Expected Shortfall, and Value-at-Risk. For the latter, our elicitation procedure initially fails due to the triviality of supporting set extrema. We therefore develop a suitable modification.
- [9] arXiv:2507.13829 (交叉列表自 eess.SP) [中文pdf, pdf, html, 其他]
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标题： 关于噪声信号谱图零点的两个基本性质标题： On two fundamental properties of the zeros of spectrograms of noisy signals主题： 信号处理 (eess.SP) ; 概率 (math.PR)
当信号被添加到白高斯噪声中时，频谱图零点的空间分布会发生显著变化。 零点倾向于勾勒出信号的支撑区域，在干扰存在的情况下会形成确定性结构，仿佛零点被困住了一样。 虽然已经提出了复杂的方法来检测频谱图零点模式中的信号作为孔洞，但很少有正式的论证来支持勾勒和被困效应。 通过针对简单玩具信号的详细计算，我们表明两个基本的数学论证，即零点的强度和儒歇定理，可以用来讨论勾勒和被困现象，以及信噪比等参数的影响。 特别是，干扰的线性调频信号，即使几乎叠加在一起，也会在零点之间产生易于检测的确定性结构。
The spatial distribution of the zeros of the spectrogram is significantly altered when a signal is added to white Gaussian noise. The zeros tend to delineate the support of the signal, and deterministic structures form in the presence of interference, as if the zeros were trapped. While sophisticated methods have been proposed to detect signals as holes in the pattern of spectrogram zeros, few formal arguments have been made to support the delineation and trapping effects. Through detailed computations for simple toy signals, we show that two basic mathematical arguments, the intensity of zeros and Rouch\'e's theorem, allow discussing delineation and trapping, and the influence of parameters like the signal-to-noise ratio. In particular, interfering chirps, even nearly superimposed, yield an easy-to-detect deterministic structure among zeros.
- [10] arXiv:2507.14058 (交叉列表自 math.AP) [中文pdf, pdf, 其他]
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标题： 适定性与多智能体模型中策略和扩散效应的混沌传播标题： Well posedness and propagation of chaos for multi-agent models with strategies and diffusive effects主题： 偏微分方程分析 (math.AP) ; 概率 (math.PR)
一个针对具有策略且受扩散效应影响的个体的多智能体模型被提出。 每个智能体的微观状态由空间位置和在紧致度量空间上的概率测度描述，该测度被解释为混合策略。 演化由非局部相互作用机制和作用于状态空间部分的随机效应所支配。 证明了多智能体系统的适定性和某种McKean--Vlasov随机微分方程的适定性。 最终，得到了一个混沌传播结果，该结果保证当智能体数量趋于无穷时，前一模型收敛到后一模型。
A multi-agent model for individuals endowed with strategies and subject to diffusive effects is proposed. The microscopic state of each agent is described by a spatial position and a probability measure, interpreted as a mixed strategy, over a compact metric space. The evolution is governed by a non-local interaction mechanism and by stochastic effects acting on the spatial component of the state. The well-posedness of the multi-agent system and that of a certain McKean--Vlasov stochastic differential equation are proved. Eventually, a propagation of chaos result is obtained, which guarantees that the former model converges to the latter as the number of agents goes to infinity.
交叉提交 (展示 4 之 4 条目 )
- [11] arXiv:2211.11463 (替换) [中文pdf, pdf, html, 其他]
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标题： 均匀多组分Curie-Weiss-Potts模型的动力学相变标题： Dynamical phase transition for the homogeneous multi-component Curie-Weiss-Potts model主题： 概率 (math.PR) ; 统计力学 (cond-mat.stat-mech) ; 数学物理 (math-ph)
在本文中，我们研究具有$q \geq 3$自旋的齐次多组分 Curie-Weiss-Potts 模型。该模型定义在完全图$K_{Nm}$上，其顶点集被等分为$m$个大小为$N$的组件。 对于配置$\sigma: \{1, \cdots, Nm\} \to \{1, \cdots, q\},$，吉布斯测度由$$ \mu_{N,\beta}(\sigma) =\frac{1}{Z_{N,\beta}} \exp\Big(\frac{\beta}{N} \sum_{v,w=1}^{Nm}\mathcal{J}(v,w)\, \mathbb{1}_{\{\sigma(v)=\sigma(w)\}}\Big), $$定义，其中$Z_{N, \beta}$是一个归一化常数，$\beta>0$是逆温度参数。 相互作用系数为$ \mathcal{J}(v, w) = \frac{J}{1 + (m-1) \lambda}$，对于同一组分中的$v, w$，以及$\mathcal{J}(v, w) = \frac{J \lambda}{1 + (m-1)\lambda}$对于不同组分中的$v, w$，其中$\lambda \in (0, 1)$是组间相互作用相对于组内相互作用的相对强度，$J>0$是有效相互作用强度。 我们识别出在临界逆温度$\beta_{\operatorname{cr}} = \beta_{s}(q)/J$处的动态相变，其中$\beta_{s}(q)$是保证自由能在Curie-Weiss-Potts模型中存在唯一临界点的最大逆温度 arXiv:1204.4503。 通过将聚合路径方法 arXiv:1312.6728 扩展到我们的多组分设置，我们证明了在高温区域$\beta<\beta_{s}(q)/J.$中的$O(N \log N)$混合时间。在低温区域$\beta > \beta_{s}(q)/J,$，我们进一步通过亚稳态证明了指数级混合时间。 这是关于多组分Potts模型中动态相变的第一个结果。
In this paper, we study the homogeneous multi-component Curie-Weiss-Potts model with $q \geq 3$ spins. The model is defined on the complete graph $K_{Nm}$, whose vertex set is equally partitioned into $m$ components of size $N$. For a configuration $\sigma: \{1, \cdots, Nm\} \to \{1, \cdots, q\},$ the Gibbs measure is defined by $$ \mu_{N,\beta}(\sigma) =\frac{1}{Z_{N,\beta}} \exp\Big(\frac{\beta}{N} \sum_{v,w=1}^{Nm}\mathcal{J}(v,w)\, \mathbb{1}_{\{\sigma(v)=\sigma(w)\}}\Big), $$ where $Z_{N, \beta}$ is a normalizing constant, and $\beta>0$ is the inverse temperature parameter. The interaction coefficients are $ \mathcal{J}(v, w) = \frac{J}{1 + (m-1) \lambda}$, for $v, w$ in the same component, and $\mathcal{J}(v, w) = \frac{J \lambda}{1 + (m-1)\lambda}$ for $v, w$ in the different components, where $\lambda \in (0, 1)$ is the relative strength of inter-component interaction to intra-component interaction, and $J>0$ is the effective interaction strength. We identify a dynamical phase transition at the critical inverse temperature $\beta_{\operatorname{cr}} = \beta_{s}(q)/J$, where $\beta_{s}(q)$ is maximal inverse temperature guaranteeing a unique critical point of the free energy in the Curie-Weiss-Potts model arXiv:1204.4503. By extending the aggregate path method arXiv:1312.6728 to our multi-component setting, we prove $O(N \log N)$ mixing time in the high-temperature regime $\beta<\beta_{s}(q)/J.$ In the low-temperature regime $\beta > \beta_{s}(q)/J,$ we further show exponential mixing time by a metastability. This is the first result for the dynamical phase transition in the multi-component Potts model.
- [12] arXiv:2408.01268 (替换) [中文pdf, pdf, 其他]
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标题： 谣言传播依赖于社交网络模型中的潜在几何结构和度分布标题： Rumour Spreading Depends on the Latent Geometry and Degree Distribution in Social Network Models评论： 49页主题： 概率 (math.PR) ; 社会与信息网络 (cs.SI) ; 组合数学 (math.CO)
我们研究在社交网络的超小世界模型中推拉谣言传播，其中度数遵循幂律分布。 在非几何设置中，Fountoulakis、Panagiotou 和 Sauerwald 证明了谣言总是以双对数时间（SODA 2012）超快速传播。 另一方面，Janssen 和 Mehrabian 发现在空间优先连接模型中，谣言传播缓慢（多项式时间）（SIDMA 2017）。 我们系统地研究了几何非均匀随机图（GIRGs）模型中的这一问题。 我们的结果分为两部分：首先，在欧几里得几何中，根据幂律指数和网络中几何强度的不同，可能产生缓慢、快速（对数多项式）或超快速谣言传播，并且我们完全表征了它们之间的相位边界。 这些区域并不与图距离区域一致，即即使图距离是双对数的，也可能出现对数多项式甚至多项式谣言传播。 我们预计这些结果可以轻松适用于相关模型，例如尺度无序渗透。 其次，我们证明在非度量几何中，谣言传播总是（至少）快速的。 所考虑的非度量几何可以用来建模社会联系，其中顶点在单一属性（如家庭血缘）上的相似性已经强烈表明边的存在。 欧几里得几何无法捕捉此类关系。 在欧几里得设置的一些区域中，传播谣言的有效路径不同于之前识别的路径。 例如，一个度数为$d$的顶点可以通过长度为$3$的链将谣言传递给一个度数更大的顶点，其中两个中间顶点中的一个是常数度，另一个的度数为$d^{c}$，其中$c<1$是某个常数。 类似的但更长的顶点链，所有顶点的度数都不是常数，也被证明是有用的。
We study push-pull rumour spreading in ultra-small-world models for social networks where the degrees follow a power-law distribution. In a non-geometric setting, Fountoulakis, Panagiotou and Sauerwald have shown that rumours always spread ultra-fast (SODA 2012), i.e. in doubly logarithmic time. On the other hand, Janssen and Mehrabian have found that rumours spread slowly (polynomial time) in a spatial preferential attachment model (SIDMA 2017). We study the question systematically for the model of Geometric Inhomogeneous Random Graphs (GIRGs). Our results are two-fold: first, with Euclidean geometry slow, fast (polylogarithmic) and ultra-fast rumour spreading may occur, depending on the exponent of the power law and the strength of the geometry in the networks, and we fully characterise the phase boundaries in between. The regimes do not coincide with the graph distance regimes, i.e., polylogarithmic or even polynomial rumour spreading may occur even if graph distances are doubly logarithmic. We expect these results to hold with little effort for related models, e.g. Scale-Free Percolation. Second, we show that rumour spreading is always (at least) fast in a non-metric geometry. The considered non-metric geometry allows to model social connections where resemblance of vertices in a single attribute, such as familial kinship, already strongly indicates the presence of an edge. Euclidean geometry fails to capture such ties. For some regimes in the Euclidean setting, the efficient pathways for spreading rumours differ from previously identified paths. For example, a vertex of degree $d$ can transmit the rumour to a vertex of larger degree by a chain of length $3$, where one of the two intermediaries has constant degree, and the other has degree $d^{c}$ for some constant $c<1$. Similar but longer chains of vertices, all having non-constant degree, turn out to be useful as well.
- [13] arXiv:2501.15633 (替换) [中文pdf, pdf, html, 其他]
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标题： 逐次遍历定理和埃德斯-雷尼的大数定律标题： Iterated Ergodic Theorems and Erd\" os--R\' enyi law of large numbers评论： 13页主题： 概率 (math.PR)
我们得到多重迭代和与积分的遍历定理，形式为$\Sigma^{(\nu)}(t)=\sum_{0\leq k_1<...<k_\nu\leq t}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$，$t\in[0,T]$和$\Sigma^{(\nu)}(t)=\int_{0\leq s_1\leq...\leq s_\nu\leq t}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$，其中$\{\xi(k)\}_{-\infty<k<\infty}$和$\{\xi(s)\}_{-\infty<s<\infty}$是向量过程，对于这些过程标准遍历定理成立，即当$\nu=1$成立时。最后我们还证明了迭代和与积分的Erdös--Rënyi大数定律的一个版本。
We obtain ergodic theorems for multiple iterated sums and integrals of the form $\Sigma^{(\nu)}(t)=\sum_{0\leq k_1<...<k_\nu\leq t}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and $\Sigma^{(\nu)}(t)=\int_{0\leq s_1\leq...\leq s_\nu\leq t}\xi(s_1)\otimes\cdots\otimes\xi(s_\nu)ds_1\cdots ds_\nu$ where $\{\xi(k)\}_{-\infty<k<\infty}$ and $\{\xi(s)\}_{-\infty<s<\infty}$ are vector processes for which standard ergodic theorems, i.e. when $\nu=1$, hold true. At the end we prove also a version of the Erd\" os--R\" enyi law of large numbers for iterated sums and integrals.
- [14] arXiv:2503.10859 (替换) [中文pdf, pdf, 其他]
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标题： 分数阶 Sobolev 过程在 Wasserstein 空间及其能量最小化粒子表示与应用标题： Fractional Sobolev processes on Wasserstein spaces and their energy-minimizing particle representations with applications评论： 40页，4图，1表主题： 概率 (math.PR) ; 优化与控制 (math.OC)
给定一个概率测度值过程 $(\mu_t)$，我们的目标是在所有一维时间边际几乎必然与 $(\mu_t)$（若存在）重合的路径连续随机过程中，找到一个最小化给定能量期望的过程。 基于我们最近的研究 (arXiv:2502.12068)，其中研究了 Wasserstein 空间中确定性路径的分数阶 Sobolev 能量最小化问题，现在我们将结果扩展到随机场景，以解决最初激发我们研究的一些应用。 给出了两个应用。 我们利用最优传输，构建了 $\mathbb{R}$ 上 Wasserstein 空间中具有 Hölder 正则性的最小化过程的粒子表示。 我们利用 Lacker--Shkolnikov--Zhang 的随机叠加原理 (J. Eur. Math. Soc. 25, 3229--3288 (2023)) 证明了满足可积条件的 $\mathbb{R}^\mathrm{d}$ 上随机 Fokker--Planck--Kolmogorov 方程解的最小化粒子表示的存在性。
Given a probability-measure-valued process $(\mu_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide almost surely with $(\mu_t)$ (if there is any), a process that minimizes a given energy in expectation. Building on our recent study (arXiv:2502.12068), where the minimization of fractional Sobolev energy was investigated for deterministic paths on Wasserstein spaces, we now extend the results to the stochastic setting to address some applications that originally motivated our study. Two applications are given. We construct minimizing particle representations for processes on Wasserstein spaces on $\mathbb{R}$ with H\"{o}lder regularity, using optimal transportation. We prove the existence of minimizing particle representations for solutions to stochastic Fokker--Planck--Kolmogorov equations on $\mathbb{R}^\mathrm{d}$ satisfying an integrability condition, using the stochastic superposition principle of Lacker--Shkolnikov--Zhang (J. Eur. Math. Soc. 25, 3229--3288 (2023)).
- [15] arXiv:2502.13529 (替换) [中文pdf, pdf, 其他]
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标题： 非均匀扩张映射类的混合性质。 应用到H{ö}哈德里安不变性原理标题： Mixing properties of a class of nonuniformly expanding maps. Application to H{ö}lderian invariance principlesAurélie Bigot (LAMA), V Alouin (ENS de Lyon)主题： 动力系统 (math.DS) ; 概率 (math.PR)
我们研究当返回到基的次数具有阶为p>1的弱矩（考虑一个缓慢变化的函数）时，一类非均匀扩张映射的混合性质。 从这些计算中，我们推导出在Hölder空间中，对于Hölder连续观测函数的Birkhoff和的部分和过程的不变性原理。 这些结果适用于单位区间的一类间歇映射。 对于这样的映射，我们也证明Hölder不变性原理对BV观测函数仍然成立。
We study the mixing properties of a class of nonuniformly expanding maps when the return time to the basis has a weak moment of order p >1, up to a slowly varying function. From these computations, we deduce an invariance principle in H{\"o}lder spaces for the partial sum process of Birkhoff sums of H{\"o}lder continuous observables. The results apply to a class of intermittent maps of the unit interval. For such a map, we also prove that the H{\"o}lder invariance principle remains true for BV observables.
- [16] arXiv:2504.18292 (替换) [中文pdf, pdf, html, 其他]
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标题： 区间上KPZ方程的大时间累积量标题： Large time cumulants of the KPZ equation on an interval评论： 38页 + 附录7页。v2：增加了参考文献。v3：回应了审稿人的意见。主题： 数学物理 (math-ph) ; 统计力学 (cond-mat.stat-mech) ; 概率 (math.PR)
我们考虑区间$[0,L]$上的 Kardar-Parisi-Zhang 方程，带有 Neumann 类边界条件和边界参数$u,v$。我们证明高度的$k$阶累积量在大时间极限$t \to +\infty$下表现为$c_k(L,u,v)\, t$，并计算了系数$c_k(L,u,v)$。我们得到了高度上尾大偏差函数的表达式。 我们还考虑大$L$的极限，其中$u=\tilde u/\sqrt{L}$，$u=\tilde v/\sqrt{L}$，这应该为区间上的两个参数族$(\tilde u, \tilde v)$KPZ 固定点给出相同的量。 我们采用两种互补的方法。 一方面，我们将 Brunet 和 Derrida 为周期情况首创的复本 Bethe 假设方法应用于区间。 另一方面，我们使用之前对开放 ASEP 可用的结果进行缩放极限。 后一种方法允许将 KPZ 方程的累积量表示为涉及积分算子的泛函方程。
We consider the Kardar-Parisi-Zhang equation on the interval $[0,L]$ with Neumann type boundary conditions and boundary parameters $u,v$. We show that the $k$-th order cumulant of the height behaves as $c_k(L,u,v)\, t$ in the large time limit $t \to +\infty$, and we compute the coefficients $c_k(L,u,v)$. We obtain an expression for the upper tail large deviation function of the height. We also consider the limit of large $L$, with $u=\tilde u/\sqrt{L}$, $u=\tilde v/\sqrt{L}$, which should give the same quantities for the two parameter family $(\tilde u, \tilde v)$ KPZ fixed point on the interval. We employ two complementary methods. On the one hand we adapt to the interval the replica Bethe ansatz method pioneered by Brunet and Derrida for the periodic case. On the other hand, we perform a scaling limit using previous results available for the open ASEP. The latter method allows to express the cumulants of the KPZ equation in terms a functional equation involving an integral operator.
- [17] arXiv:2505.18879 (替换) [中文pdf, pdf, 其他]
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标题： 通过随机性再利用的高效在线随机抽样标题： Efficient Online Random Sampling via Randomness Recycling评论： 35页，9图，2表，14算法主题： 数据结构与算法 (cs.DS) ; 离散数学 (cs.DM) ; 信息论 (cs.IT) ; 概率 (math.PR) ; 计算 (stat.CO)
“随机性再利用”是一种强大的算法技术，用于重新使用概率算法消耗的随机信息的一部分，以减少其熵需求。 本文提出了一类随机性再利用算法，用于高效采样一个服从任意随机过程的离散随机变量序列$X_1, X_2, X_3, \dots$。 我们开发了随机性再利用技术，以降低多种著名采样算法的熵成本，这些算法包括均匀采样、逆变换采样、查找表采样、别名采样和离散分布生成（DDG）树采样。 我们的方法在使用$O(\log(1/\varepsilon))$空间时，每输出样本的期望摊还熵成本为$H(X_1,\dots,X_k)/k + \varepsilon$输入位，这与$k\to\infty$的最优香农熵率$H(X_1,\dots,X_k)/k$位每样本非常接近。 我们方法的空间、时间和熵特性相结合，优于Knuth和Yao的熵最优算法以及Han和Hoshi的区间算法，用于采样离散随机序列。 在实验方面，我们展示了当使用密码学安全的伪随机数生成器时，随机性再利用能够实现Fisher-Yates洗牌的最先进运行时性能；它还可以加速离散高斯采样器。 随文附带了一个高性能的C语言软件库，该库使用随机性再利用来加速几种现有的随机采样算法。
``Randomness recycling'' is a powerful algorithmic technique for reusing a fraction of the random information consumed by a probabilistic algorithm to reduce its entropy requirements. This article presents a family of randomness recycling algorithms for efficiently sampling a sequence $X_1, X_2, X_3, \dots$ of discrete random variables whose joint distribution follows an arbitrary stochastic process. We develop randomness recycling techniques to reduce the entropy cost of a variety of prominent sampling algorithms, which include uniform sampling, inverse transform sampling, lookup-table sampling, alias sampling, and discrete distribution generating (DDG) tree sampling. Our method achieves an expected amortized entropy cost of $H(X_1,\dots,X_k)/k + \varepsilon$ input bits per output sample using $O(\log(1/\varepsilon))$ space as $k\to\infty$, which is arbitrarily close to the optimal Shannon entropy rate of $H(X_1,\dots,X_k)/k$ bits per sample. The combination of space, time, and entropy properties of our method improves upon the Knuth and Yao entropy-optimal algorithm and Han and Hoshi interval algorithm for sampling a discrete random sequence. On the empirical side, we show that randomness recycling enables state-of-the-art runtime performance on the Fisher-Yates shuffle when using a cryptographically secure pseudorandom number generator; and it can also speed up discrete Gaussian samplers. Accompanying the manuscript is a performant software library in the C programming language that uses randomness recycling to accelerate several existing algorithms for random sampling.
- [18] arXiv:2507.05959 (替换) [中文pdf, pdf, html, 其他]
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标题： 环面部分双曲自映射的极限定理标题： Limit theorems for toral partially hyperbolic endomorphisms主题： 动力系统 (math.DS) ; 概率 (math.PR)
在对可观测量的自然假设下，我们证明了对于二维环面的一类广泛的部分双曲自映射，存在中心极限定理、Berry-Esseen定理以及定量局部极限定理。我们的结果适用，但不仅限于斜积及其扰动，并且即使系统存在多个但有限多个绝对连续的遍历不变测度时，它们仍然有效。
Under natural assumptions on the observable, we prove a Central Limit Theorem, a Berry-Esseen Theorem, and a quantitative Local Limit Theorem for a broad class of partially hyperbolic endomorphisms of the two-dimensional torus. Our results apply, but are not limited to, skew-products and their perturbations, and they remain valid even when the system admits multiple, though finitely many, absolutely continuous ergodic invariant measures.