PERCOLATION THEORY

A TCC COURSE HELD FOR GRADUATE STUDENTS OF MATHEMATICS AT BATH, BRISTOL, IMPERIAL COLLEGE LONDON, OXFORD, SWANSEA AND WARWICK

2022 AUTUMN (OXFORD: MICHAELMAS) TERM

LECTURER: BÁLINT TÓTH (BRISTOL AND BUDAPEST)

WHEN?

LECTURES HELD ON WEDNESDAYS 10:00-12:00 (UK TIME)

FIRST LECTURE: WEDNESDAY, 12 OCTOBER 2022

LAST LECTURE: WEDNESDAY, 30 NOVEMBER 2022

WHERE?

THE LECTURES WILL BE HELD ON MICROSOFT TEAMS PLATFORM WITH ACCESS PROVIDED TO REGISTERED ATTENDANTS.

MESSAGES TO STUDENTS ATTENDING THE COURSE:

12 DECEMBER 2022:

• DEADLINE OF REPORTS/ESSAYS: 25 JANUARY 2023.

GENERAL OVERVIEW:

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.

Prerequisites:

• • • 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.

LECTURE NOTES (HAND WRITTEN):

1. INTRODUCTION AND BASICS: BASIC COUPLING, HARRIS INEQUALITY, SUBADDITIVE CONVERGENCE, CONSEQUENCES

2. SUPERCRITICAL PERCOLATION: CONTINUITY OF THE PERCOLATION PROBABILITY AND UNIQUENESS OF THE INFINITE CLUSTER

3. FURTHER TECHNICAL TOOLS: VAN DEN BERG-KESTEN INEQUALITY AND RUSSO'S FORMULA

4. EXPONENTIAL DECAY IN SUBCRITICAL PERCOLATION AND SHARPNESS OF THE PHASE TRANSITION

5. 2-DIMENSIONS, 1: PLANAR DUALITY AND CONSEQUENCES. HARRIS-KESTEN-RUSSO THEOREM STREAMLINED

6. 2-DIMENSIONS, 2: RUSSO-SEYMOUR-WELSH THEOREM AND CONSEQUENCES

7. 2-DIMENSIONS, 3: CONFORMAL INVARIANCE OF 2-D CRITICAL PERCOLATION -- CARDY'S FORMULA AND SMIRNOV'S THEOREM

8. SLE (STOCHASTIC/SCHRAMM-LÖWNER EVOLUTION) -- A LIGHT INTRODUCTION

ASSESSMENT FOR CREDIT:

Those PhD students who take this unit for credit will be assessed in the following way: I will provide some extra material for reading. This will be either a research paper or some theoretical part not fully covered in class. The student will be asked to read and understand the material and to write an essay of cca 15 pages about it, in the style and format of a research paper.

Timing: The reading material will be provided towards the end of the TCC term (by end of November or early December). Deadline for handing in the essay sometime in the winter break (mid- or end of January).

Please let me know whether you take this unit for credit or for audit only.

FURTHER READING (BOOKS):

• 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

link to the book's web page at CUP

FURTHER READING (PAPERS)

There are dozens of papers published yearly in the field of mathematical percolation theory. Some of paramount importance will be mentioned during the lectures.