| Abstract: | We prove that for the critical level-set of Gaussian free field on the metric graph generated by Z^d (d>6), the one-arm probability (i.e. the probability of the event that the origin is connected to the boundary of the box B(N)) is proportional to N^{-2}, where B(N) is centered at the origin and has side length 2N. This paper is a joint work with Prof. Jian Ding. In addition to presenting the proof idea, this talk will also involve some interesting open questions, including the one-arm exponent in low dimensions, the structure of large clusters, and the incipient infinite cluster. |
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