Tom Hutchcroft: Critical cluster volumes in hierarchical percolation


Hierarchical percolation is a toy model for percolation on Z^d that is expected to exhibit much of the same critical phenomena as percolation on Z^d, including having mean-field behaviour in high dimensions, non-mean-field behaviour with dimension-dependent exponents in low dimensions, and logarithmic corrections to mean-field behaviour at the upper-critical dimension. In this talk I will discuss my recent preprint in which a complete proof of this critical behaviour is established for the distribution of cluster volumes in all dimensions. In particular, I will discuss how one can implement what is in effect a rigorous Wilson-style renormalization group argument directly for percolation, without needing to reduce to some equivalent spin system. (Indeed, no such spin system is known to exist for percolation!) No previous experience with the renormalization group will be assumed.




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