分支随机游走的小球事件重整化
Renormalizing small ball events for branching random walk
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中文总结 AI 辅助
该研究通过重整化群方法证明，受小球事件条件作用的分支随机游走端点间相关性指数衰减，相关长度为最优的 O(r)，并推广至 sinh-Gordon 和广义 φ^4 模型。
中文摘要 AI 辅助
考虑深度为 $n$ 的分支随机游走（BRW），其增量为标准高斯分布，并限制所有端点位于半径为 $r$（可能依赖于 $n$）的区间内，即一个小球事件。我们证明这种条件作用导致端点之间的相关性指数衰减，相关长度为 $O(r)$。我们的证明基于重整化群，在此背景下，重整化群捕捉了端点条件作用对游走早期步骤的影响。这种影响可分为所谓的相关部分和无关部分，在我们的设定中，相关部分使得条件化的分支随机游走在最初 $n-O(r)$ 步内表现得像分支 Ornstein-Uhlenbeck 游走，从而导致相关性衰减。无关部分以非平凡的方式收缩，而最简单的统一界定它的论证仅产生相关长度 $O(r^2)$。为了实现最优的 $O(r)$ 界，我们证明无关部分在空间上水平收缩后才均匀衰减。分支随机游走是对数相关高斯场的典型例子，并且它构成了量子场论数学表述中许多层级场论的基础。因此，我们以一般形式呈现我们的结果，该形式适用于其他具有约束势的层级场论，其中小球条件化的分支随机游走就是一个例子。我们将一般结果应用于另外两个模型：第一，sinh-Gordon 模型，其势函数严格凸，因此在协方差层面，该模型表现得像具有二次势的模型，即模型具有正质量。第二，我们考虑 $\u03c6^4$ 模型的推广，其幂次 $\u03b1 \geq 2$；随着 $\u03b1$ 从 2 增加到 $\infty$，这些幂次势自然地在有质量（二次）势和小球事件的无限深方势阱之间插值。
英文摘要
Consider a branching random walk (BRW) of depth $n$ with standard Gaussian increments, conditioned on all endpoints lying in an interval of radius $r$ (which may depend on $n$), i.e. a small ball event. We prove that this conditioning results in exponential decay of correlations between the endpoints, with correlation length $O(r)$. Our proof is based on the renormalization group, which in this context captures the effect of the endpoint conditioning on earlier steps in the walk. This effect can be separated into so-called relevant and irrelevant parts, and in our setting the relevant part makes the conditioned BRW behave like a branching Ornstein-Uhlenbeck walk for the first $n-O(r)$ steps, leading to the correlation decay. The irrelevant part shrinks in a nontrivial manner, and the simplest argument to bound it uniformly only yields correlation length $O(r^2)$. To achieve the optimal $O(r)$ bound, we establish that the irrelevant part shrinks horizontally in space before decaying uniformly. The BRW is a prototypical example of a log-correlated Gaussian field, and it forms the basis of many hierarchical field theories in the mathematical formulation of quantum field theory. As such, we present our results in a general form which is applicable to other hierarchical field theories with confining potentials, of which the small ball conditioned BRW is an example. We apply our general result to two other models: first, the sinh-Gordon model which has a strictly convex potential, so at the level of covariances the model behaves like one with a quadratic potential, i.e. the model has a positive mass. Second, we consider generalizations of the $ϕ^4$ model with any power $α\geq 2$; as $α$ increases from $2$ to $\infty$, these power potentials naturally interpolate between the massive (quadratic) potential and the infinite square well potential of the small ball event.
