Colloquium / Seminars

Topic：Critical point for oriented percolation
Speaker：Dr. Noe Kawamoto
　　　　(NCTS)
Date time：April 1, 2025 14:00 - 15:00
Venue：SA213
Abstract：
Abstract. We consider nearest-neighbor oriented percolation defined on the product space of a multi-dimensional integer lattice and the set of positive integers.

A point in the product space is described by a vector, where the first component (space component) is a point of the lattice and the second component (time component) is a positive integer.

For a pair of points, where the space components are neighbors and the difference in their time component is 1, we can define a bond, which is independently open with probability p/2d with 0 ≤ p ≤ 2d, regardless of the other bonds. It is well known that oriented percolation exhibits a phase transition as the parameter p varies around a critical point pc which is model-dependent. As the dimension tends to infinity, pc coverges to 1.

However, the best estimate for pc provided by Cox and Durret (Math. Proc. Camb. Phil. Soc. (1983)) give upper and lower bounds, but do not yield an explicit expression for pc.

In this talk, we investigate the explicit expression for pc when d > 4, in a way that pc = 1 + C1d^{-2} + C2d^{-3} + C3d^{-4}+ O(d−5), where C1 to C3 are constants. The proof relies on the lace expansion, which is one of the most powerful tool to analyze the mean-field behavior of statistical-mechanical models in high dimensions. We focus less on the details of the proof and more on the background related to the topic.

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