A pattern theorem for lattice clusters

A Cluster Limit Theorem for Infinitely Divisible Point Processes

In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in distribution of $(N_n)_{n \geq 1}$ to an infinitely divisible point process $N$. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.

Central limit theorems in Random cluster and Potts Models

We prove that for q>=1, there exists r(q)<1 such that for p>r(q), the number of points in large boxes which belongs to the infinite cluster has a normal central limit behaviour under the random cluster measure phi_{p,q} on Z^d, d>=2. Particularly, we can take r(q)=p_g^* for d=2, which is commonly conjectured to be equal to p_c. These results are used to prove a q-dimensional central limit theorems relative to the fluctuation of the empirical measures for the ground Gibbs measures of the q-state Potts model at very low temperature and the Gibbs measures which reside in the convex hull of them. …

Cluster capacity functionals and isomorphism theorems for Gaussian free fields

We investigate level sets of the Gaussian free field on continuous transient metric graphs $\tilde{\mathcal G}$ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $\mathcal{G}$, that the capacity of sign clusters on $\tilde{\mathcal G}$ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $\tilde{\mathcal G}$ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $\tilde{\mathcal G}$ vanishes in most instances in the m…

连续渗流簇中反射扩散的局部中心极限定理

Local Central Limit Theorem for Reflecting Diffusions in a Continuum Percolation Cluster

本文研究连续渗流簇上的反射扩散问题，在无界簇唯一且满足若干几何条件的假设下，证明了对应的淬火局部中心极限定理。

固定点偏向避免模式的置换的极限定理

Limit Theorems for Fixed Point Biased Pattern Avoiding Involutions

本文研究了避免特定模式的固定点偏向置换，通过分析不同模式的偏置强度，揭示了固定点数量的渐近分布特性。

Central limit theorems for patterns in multiset permutations and set partitions

We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of size 2 in both settings, obtained by Canfield, Janson and Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.

Parabolic Harnack Inequality and Local Limit Theorem for Percolation Clusters

We consider the random walk on supercritical percolation clusters in the d-dimensional Euclidean lattice. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.

Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields

We study occurrences of patterns on clusters of size n in random fields on Z^d. We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most an times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities tha…

重复模式的一个中心极限定理

A Central Limit Theorem for Repeating Patterns

本文证明了随机排列中遵循一类重复模式的最长子序列长度的中心极限定理，指出在长排列极限下其收敛于均值和方差与排列长度呈线性关系的正态分布。

Correction note to "Limit Theorems for Empirical Processes of Cluster Functionals" [arXiv:0910.0343]

We correct an error in a technical lemma of Drees and Rootzén (2010) [arXiv:0910.0343] and discuss consequences for applications.

Limit Theorems for Patterns in Ranked Tree-Child Networks

We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of height $1$ and $2$, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to $0$ and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern.

关于避免三元模式的固定点偏向排列的极限定理

Limit theorems for fixed point biased permutations avoiding a pattern of length three

本文研究了在特定偏分布下，避免三元模式排列中固定点数量的极限行为，揭示了偏参数变化时分布形态的相变特性。

Vincular排列模式的中心极限定理

A Central Limit Theorem for Vincular Permutation Patterns

本文证明任意固定Vincular排列模式在均匀随机排列中的出现次数渐近正态，并给出收敛速度，通过依赖图方法及基于全方差定律的递归技术估计方差下界。

A pattern theorem for random sorting networks

A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random n-element sorting network, any fixed pattern occurs in at least cn^2 disjoint space-time locations, with probability tending to 1 exponentially fast as n tends to infinity. Here c is a positive constant which depends on the choice of pattern. As a consequence, the probability that the uniformly random sorting network is geometricall…

A dependent Lindeberg central limit theorem for cluster functionals on stationary random fields

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes $(Z_n(f))_{f\in\mathcal{F}}$ whose index set $\mathcal{F}$ is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depends mainly on the usual Lindeberg condition and a sequence $T_n$ which summarizes the dependence between the blocks of the random field values. Finally, as application, we use the previous result in order to show the Gaussian asymptotic behavior of the iso-extremogram esti…