无限箱模型中的冻结现象

二元感知器模型中的强冻结现象

Strong freezing in binary perceptron models

本文证明了二元感知器模型在任意固定正约束密度和边际下以高概率表现出强冻结现象，即典型解相互孤立且汉明距离为n阶，方法基于植入模型和受限配分函数，并利用FKG不等式完成证明。

向量值高斯自由场避免球体的条件：范数的熵排斥与自旋冻结

Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins

本文研究了二维向量值狄利克雷高斯自由场及其质量化晶格对应物，在避免子域内每个点的球体的条件下，证明了无质量场的范数表现出熵排斥，而其角分量在所有介观尺度上冻结。

聚合物钉扎动力学的标度极限和一维Stefan冻结问题

The scaling limit of polymer pinning dynamics and a one dimensional Stefan freezing problem

我们考虑了1+1维界面（或聚合物）在基底存在下的随机演化。这个随机过程是均匀钉扎模型的动力学版本。我们从一个远离平衡的配置开始：一个具有非平凡宏观高度分布的聚合物，并研究时空尺度界面的演化。在两种情况下，我们证明了这种重标度的界面在扩散尺度上具有标度极限（在两个维度上由$L$重新缩放的空间和由$L^2$重新缩放的时间，其中$L$表示接口的长度），我们描述：当与衬底的相互作用使得系统在平衡时未被钉扎时，然后通过求解Dirichlet边界条件下的热传导方程，给出了高度剖面的标度极限;当对基底的吸引力为无穷大时，标度极限是一个自由边界问题，它属于具有收缩边界的Stefan问题类，也被称为斯特凡冻结问题。此外，我们证明了该问题的解的存在性和正则性，直到一个最大的时间，在那里的边界碰撞。

无限二项式Galton-Watson树的随机序

Stochastic Ordering of Infinite Binomial Galton-Watson Trees

我们考虑后代服从${\rm Bin}(d,p)$分布的Galton-Watson树。我们让$T_{\infty}(p)$表示这样的树条件是无限的。对于$d=2,3$和任何$1/d\leq p_1 <p_2 \leq 1$，我们证明了$T_{\infty}(p_1)$和$T_{\infty}(p_2)$之间存在耦合，使得${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$

Coupling any number of balls in the infinite-bin model

The infinite-bin model, introduced by Foss and Konstantopoulos, describes the Markovian evolution of configurations of balls placed inside bins, obeying certain transition rules. We prove that we can couple the behaviour of any finite number of balls, provided at least two different transition rules are allowed. This coupling makes it possible to define the regeneration events needed by Foss and Zachary to prove convergence results for the distribution of the balls.

Two-sided infinite-bin models and analyticity for Barak-Erdős graphs

In this article, we prove that for any probability distribution $μ$ on $\mathbb{N}$ one can construct a two-sided stationary version of the infinite-bin model (an interacting particle system introduced by Foss and Konstantopoulos) with move distribution $μ$. Using this result, we obtain a new formula for the speed of the front of infinite-bin models, as a series of positive terms. This implies that the growth rate $C(p)$ of the longest path in a Barak-Erdős graph of parameter $p$ is analytic on $(0,1]$.