2022 AUTUMN (OXFORD: MICHAELMAS) TERM
DEADLINE OF REPORTS/ESSAYS: 25 JANUARY 2023.
This is an intyroductory course about one of the most dynamically developing areas of modern probability theory, linked on many channels with various other chapters of mathematics and physics. Remarkable results achieved in this area have been recently recognised with three Fields Medals (Werner 2006, Smirnov 2010, Duminil-Copin 2022). The course will cover a wide spectrum of the theory. I will cover the following topics, providing full mathematical treatment of all main results:
Phenomenology, geometry of random graphs, phase transition.
Elementary tools: stochastic ordering and basic correlation inequalities.
Supercritical behaviour: uniqueness of percolating cluster and regularity.
Subcritical behaviour: exponential decay of connectivity and sharpness of the phase transition.
Planar percolation: topological duality and its consequences.
Conformal invariance of critical planar percolation.
Outlook.
Basic probability theory (including stochastic independence, Markov property, laws of large numbers, 0-1 laws, ergodicity)
Basic analysis (including uniform convergence, Arzelà-Ascoli theorem)
Basic complex function theory (including Riemann’s conformal mapping theorem)
However, those hesitating possible attendants who miss some of these elements should not be shy. The necessary background knowledge can be picked up on the way presuming enough basic mathematical interest and open mind.
INTRODUCTION AND BASICS: BASIC COUPLING, HARRIS INEQUALITY, SUBADDITIVE CONVERGENCE, CONSEQUENCES
FURTHER TECHNICAL TOOLS: VAN DEN BERG-KESTEN INEQUALITY AND RUSSO'S FORMULA
EXPONENTIAL DECAY IN SUBCRITICAL PERCOLATION AND SHARPNESS OF THE PHASE TRANSITION
2-DIMENSIONS, 1: PLANAR DUALITY AND CONSEQUENCES. HARRIS-KESTEN-RUSSO THEOREM STREAMLINED
2-DIMENSIONS, 2: RUSSO-SEYMOUR-WELSH THEOREM AND CONSEQUENCES
SLE (STOCHASTIC/SCHRAMM-LÖWNER EVOLUTION) -- A LIGHT INTRODUCTION
Harry Kesten: Percolation Theory for Mathematicians. Birkhauser, 1982
link to the book's web page at Springer
link to the author's web page where PDFs of chapters are freely available
Geoffrey Grimmett: Percolation. Second edition. Springer, 1999
link to the book's web page at Springer
Bela Bollobas and Oliver Riordan: Percolation. Cambridge University Press, 2006
There are dozens of papers published yearly in the field of mathematical percolation theory. Some of paramount importance will be mentioned during the lectures.