Mathematical Study of Critical Phenomena for Statistical Models in Probability

研究的主要目的是探讨概率论和统计力学中出现的某些随机几何模型的临界行为。特别地，我们研究了高维临界渗流模型的初始无限簇的连续统（缩放）极限。我们发现，对于大小为N的簇，如果将空间按N的四次方根比例缩放，可以得到一个合理的连续统极限。此外，我们计算了极限分布的第一和第二矩，并将它们识别为积分超布朗偏移（ISE）的第一和第二矩。这有力地表明连续体极限实际上是ISE。证明方法采用了蕾丝展开法，这种方法在其他情况下已经得到了成功的应用。为了研究本研究的特殊问题，我们计算了聚类分布的生成函数，并应用了陶伯利分析。

The main purpose of the research was to investigate critical behavior of certain stochastic geometric models that appear in probability theory and statistical mechanics. In particular, we investigated continuum (scaling) limits of incipient infinite clusters of the critical percolation model in high dimensions.We found that a reasonable continuum limit could be obtained for a cluster of size N, if we scale the space proportional to the fourth root of N.Moreover, we calculated the first and second moments of the limiting distributions, and identified them as the first and second moments of the integrated super-Brownian excursion (ISE). This strongly suggests that the continuum limit is in fact ISE.The method of proof uses the lace expansion, which has been used successfully in other contexts. To investigate the particular problem of this research, we calculated generating functions of cluster distributions, and applied Tauberian analysis.