Abstract:
A major breakthrough in percolation was the 1990 result by Hara and Slade proving mean-field behavior of percolation in high-dimensions, showing that at criticality there is no percolation and identifying several percolation critical exponents. The main technique used is the lace expansion, a perturbation technique that allowed Hara and Slade to compare percolation paths to random walks based on the idea that faraway pieces of percolation paths are almost independent in high dimensions. In this talk, we describe these seminal 1990 results, as well as a number of novel results have appeared for high-dimensional percolation that have been derived since and that build on the shoulders of these giants.
Time permitting, I intend to highlight the following topics:
(1) Critical percolation on the tree and critical branching random walk to fix ideas and to obtain insight in the kind of results that can be proved in high-dimensional percolation;
(2) The recent computer-assisted proof, with Robert Fitzner, that identifies the critical behavior of nearest-neighbor percolation above 10 dimensions using the so-called Non-Backtracking Lace Expansion (NoBLE) that builds on the unpublished work by Hara and Slade proving mean-field behavior above 18 dimension;
(3) The identification of arm exponents in high-dimensional percolation in two works by Asaf Nachmias and Gady Kozma, using a clever and novel difference inequality argument, and its implications for the incipient infinite cluster and random walks on them;
(4) Percolation on high-dimensional tori and their finite-size corrections.
We assume no prior knowledge about percolation.
Biography:
Remco van der Hofstad received his Ph.D. at the University of Utrecht in 1997, under the supervision of Frank den Hollander and Richard Gill. Since then, he worked at McMaster University in Hamilton, Canada, and Delft Univer-sity of Technology. He is currently Full Professor in probability at Eindhoven University of Technology and Acting Scientific Director of Eurandom. Further, jointly with Frank den Hollander, he is responsible for the 'Random Spatial Structures' Program at Eurandom.
Remco works on statistical mechanics in high dimensions, on random graph models for complex networks and stochastic processes on them, and the interplay between these topics. He is further interested in exploring applications of probability in chemistry, electrical engineering and computer science with scientists from these fields.
Remco received the Prix Henri Poincaré 2003 jointly with Gordon Slade, the Rollo Davidson Prize 2007, is a laureate of the 'Innovative Research VIDI Scheme' 2003 and 'Innovative Research VICI Scheme' 2008, and was elected in the Royal Netherlands Academy of Arts and Sciences in 2018. He is also one of the 11 coapplicants of the Dutch Gravitation program NETWORKS. This 23 M euro program, a collaboration between the universities of Amsterdam, Leiden and Eindhoven, and the Center for Mathematics and Computer Science in Amsterdam, aims at combining stochastics and algorithmics to solve network problems.
Remco has lectured in most of the major summer schools in probability, including Saint-Flour 2017, Cornell 2012, PIMS-CRM in Montreal 2015, and has taught some 20 more shorter and longer courses throughout the world. He is author of some 130 papers and 2 books, with two more books in preparation. He has supervised 20 Ph.D. students, of whom 15 have completed their Ph.D., and 15 postdocs.
Remco is editor in chief of the 'Network Pages', an interactive website by the networks community for everyone interested in networks.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai