Probability and Mathematical Physics Seminar

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Evita Nestoridi (11:10 am)

Title: Shuffling via transpositions
Abstract: In their seminal work, Diaconis and Shahshahani proved that shuffling a deck of $n$ cards sufficiently well via random transpositions takes $1/2 n log n$ steps. Their argument was algebraic and relied on the combinatorics of the symmetric group. In this talk, I will focus on a generalization of random transpositions and I will discuss the underlying combinatorics for understanding their mixing behavior and indeed proving cutoff. The talk will be based on joint work with S. Arfaee.

Matthew Kahle (12:10 pm)

Title: Homological percolation in a torus
Abstract: The celebrated Harris–Kesten theorem is that the critical probability for bond percolation in the square lattice is 1/2. It has been a folklore problem for apparently some time to find an appropriate analogue of this theorem in high dimensions.
Duncan, Schweinhart, and I studied "plaquette percolation" in a d-dimensional torus made by identifying opposite sides of a subdivided d-dimensional cube, where k-dimensional cubical cells are inserted independently with probability p. For an appropriate homological definition of "giant cycles", and whenever d=2k, we show that the critical probability is indeed 1/2.