Mathematical Study of Critical Phenomena for Statistical Models in Probability

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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.