Final Report Summary - GEOMCRITRAND (The geometry of critical random and pseudorandom systems)
Probability theory is one of the fastest developing areas of mathematics, finding new connections to other branches constantly, from conformal geometry and representation theory to geometric group theory and partial differential equations. It is also indispensable in physics and computer science. A central object is percolation: the study of connected clusters in random subgraphs of a fixed underlying graph, such as the d-dimensional integer lattice. This is the simplest model exhibiting a phase transition: there is a critical density above which infinite connected components appear. The critical behaviour is always the most interesting one, and the phase transition sheds light on other statistical physics models in two and more dimensions and on Cayley graphs of groups. Percolation is also a key example of how perturbations of the underlying graph influence stochastic processes (modelling the spreading of heat or electricity in materials or of information or viruses in a population).
Gábor Pete is an expert on most aspects of percolation: conformal invariance in the plane, scaling limits, critical exponents, noise sensitivity, renormalization, strongly dependent percolation models, random walks on percolation clusters, and connections to geometric group theory. After almost a decade of successful research in the US and Canada, he returned to Europe, to the Institute of Mathematics of the Technical University of Budapest, a well-known center of probability and dynamical systems, with experts on self-interacting random walks, diffusion in disordered media, interacting particle systems, random graphs and fractals. The researcher in charge at the host institute was Bálint Tóth, a leading figure of the probability life of Hungary.
GP's proposal concerned the following three interrelated topics:
1. The advancement of two-dimensional probability by extending the understanding of critical systems to the near-critical regime and proving conformal invariance for new models.
2. Random walks (return probabilities, spectral measures, scaling limits) on groups, quasicrystals, fractal-like graphs.
3. Studying graph limits, dynamically growing graphs, and fractals, via probability and Szemerédi regularity type ideas.
Regarding transfer of knowledge, the main objectives were to disseminate ideas of conformally invariant and geometric group theoretical probability within the Hungarian mathematics community on one hand, equip GP with new tools and points of view on the other hand, and, as a joint result of these two aims, integrate GP in the mathematical life of Hungary, and establish new collaborative links worldwide.
GP's research achievements in the past two years are concentrated mostly in Topic 1: first of all, finishing projects on near-critical and dynamical percolation and related models in the plane, started with Christophe Garban (Lyon), Alan Hammond (Oxford), and the late Oded Schramm, and then, developing new ideas originating from these investigations. These earlier projects became much longer and more complex than expected, resulting in six papers (altogether 350 pages), requiring significant new ideas, involving new collaborators (Hugo Duminil-Copin (Geneva) and Elchanan Mossel (Berkeley)), and opening new areas for future research. All six papers are available freely at www.arXiv.org and have been submitted to first-class peer-reviewed journals; two of them have already appeared, one has been accepted for publication, and three of them are under review. The six papers already have 27 independent citations.
GP has also made significant progress with his book project "Probability and Geometry on Groups", lying more within Topics 2 and 3, with an increase of about 100 pages since his Marie Curie proposal, now at 220 pages. Cambridge University Press and the European Mathematical Society have shown serious interest in publishing the book.
A main outcome of the research of Garban, Pete & Schramm on percolation in the plane is that near-critical and dynamical versions of percolation can be very efficiently described using the geometry of the critical system. The methods developed are very robust and can be applied to a wide range of models related to near-critical percolation, such as understanding how exceptionally large clusters in dynamical percolation appear and how they look like (Hammond, Mossel & Pete, and Hammond, Pete & Schramm (see the attached illustration, based on animation by Vincent Beffara); what the continuum planar Minimal Spanning Tree is and what its geometric properties are (Garban, Pete & Schramm); the asymptotic circularity of the supercritical Wulff crystal of percolation (Duminil-Copin); the density of frozen sites in a percolation growth model where clusters freeze when they become large enough (Demeter Kiss (Amsterdam/Cambridge)).
The main results of Duminil-Copin, Garban & Pete are the computation of the near-critical window of the planar FK-Ising model of magnetization via conformal invariance methods and the realization that this window is determined by the critical behaviour of the system in a way very different from percolation --- refuting a natural conjecture made by several leading researchers independently. A possible consequence, work in progress sparked by a conversation with Aizenman (Princeton), is that the famous Harris criterion, predicting when the near-critical behaviour of a disordered system is the same as that of the pure system, might be wrong in many models; if our proposal is correct, it has not been discovered so far because even the way of measuring the effect of disorder has been fundamentally flawed.
Different pieces of work in progress concern the application of noise sensitivity methods developed in earlier work of Garban, Pete & Schramm to a variety of problems: universality results concerning conformal invariance of percolation (with a small result on disordered percolation, noticed first by Aizenman); proving fluctuation bounds in a famous First Passage Percolation problem; exceptional times in dynamical bootstrap percolation on trees (joint with Marek Biskup (UCLA)); the Friedgut-Kalai conjecture on Fourier Entropy versus Influence of general Boolean functions, with a new version suggested by Gábor Tardos (Budapest), possibly applicable to computing the independence ratio of random d-regular graphs (Endre Csóka (Budapest/Warwick)).
Part of the project was (jointly with Elchanan Mossel (Berkeley) and Pablo Shmerkin (Surrey)) to study arithmetic progressions and other patterns in random fractal sets on the real line. Unfortunately, serious new obstacles in the fractal-geometric approach to this problem have been identified. The more combinatorial Szemerédi-type approach is still to be studied; this will soon be undertaken together with József Balogh (Urbana-Champaign), visiting the University of Szeged as a Marie Curie fellow, and Balázs Szegedy (Toronto/Budapest).
GP has been very active in the mathematical life of Hungary. Besides teaching several graduate and undergraduate courses at the Technical University, including a very successful special topics course "Critical phenomena and conformal invariance in the plane", he has been supervising a PhD student at the Central European University (jointly with physicist János Kertész) on the topic of dynamically changing real-world networks, and three MSc and four BSc students at TU Budapest. The MSc research projects have the potential to produce publishable new results. GP has also taken part in restructuring the probability section of the PhD programme of the Central European University, Budapest, and will recurrently teach some graduate courses there. Together with Miklós Abért, he is co-organizing an extremely lively research seminar at the Rényi Institute of the Hungarian Academy of Sciences, entitled "Graphs, groups & probability", attended by many bright young researchers in Hungary. Starting at the end of his MC IIF project, GP has joined the Rényi Institute, and will remain a part-time associate professor at TU Budapest, continuing to teach important probability courses and advise students. New connections between the probability group at TU Budapest and the group theoretical-combinatorial group at the Rényi Institute are shown by a successful Hungarian Scientific Fund (OTKA) grant application, starting Sept 2013, led by Abért, with the participation of GP, Bálint Tóth, and several young researchers from both the Rényi Institute and TU Budapest.
Integration of GP into the Hungarian mathematical community is also shown by his active participation as an OTKA panel member in 2013, by receiving a 3-year Bolyai Fellowship from the Hungarian Academy of Sciences starting Sept 2013, by giving an invited talk at the Erdös Centennial, July 2013, and being a regular speaker at various probability seminars in Budapest. In a broader European context, he has joined the editorial board of the journal Annales de l'Institut Henri Poincaré, and has taught or will teach at several winter schools in Europe (Eurandom 2012, ZiF 2013, Darmstadt 2014).
In summary, the project was very successful both scientifically and regarding transfer of knowledge.
Gábor Pete is an expert on most aspects of percolation: conformal invariance in the plane, scaling limits, critical exponents, noise sensitivity, renormalization, strongly dependent percolation models, random walks on percolation clusters, and connections to geometric group theory. After almost a decade of successful research in the US and Canada, he returned to Europe, to the Institute of Mathematics of the Technical University of Budapest, a well-known center of probability and dynamical systems, with experts on self-interacting random walks, diffusion in disordered media, interacting particle systems, random graphs and fractals. The researcher in charge at the host institute was Bálint Tóth, a leading figure of the probability life of Hungary.
GP's proposal concerned the following three interrelated topics:
1. The advancement of two-dimensional probability by extending the understanding of critical systems to the near-critical regime and proving conformal invariance for new models.
2. Random walks (return probabilities, spectral measures, scaling limits) on groups, quasicrystals, fractal-like graphs.
3. Studying graph limits, dynamically growing graphs, and fractals, via probability and Szemerédi regularity type ideas.
Regarding transfer of knowledge, the main objectives were to disseminate ideas of conformally invariant and geometric group theoretical probability within the Hungarian mathematics community on one hand, equip GP with new tools and points of view on the other hand, and, as a joint result of these two aims, integrate GP in the mathematical life of Hungary, and establish new collaborative links worldwide.
GP's research achievements in the past two years are concentrated mostly in Topic 1: first of all, finishing projects on near-critical and dynamical percolation and related models in the plane, started with Christophe Garban (Lyon), Alan Hammond (Oxford), and the late Oded Schramm, and then, developing new ideas originating from these investigations. These earlier projects became much longer and more complex than expected, resulting in six papers (altogether 350 pages), requiring significant new ideas, involving new collaborators (Hugo Duminil-Copin (Geneva) and Elchanan Mossel (Berkeley)), and opening new areas for future research. All six papers are available freely at www.arXiv.org and have been submitted to first-class peer-reviewed journals; two of them have already appeared, one has been accepted for publication, and three of them are under review. The six papers already have 27 independent citations.
GP has also made significant progress with his book project "Probability and Geometry on Groups", lying more within Topics 2 and 3, with an increase of about 100 pages since his Marie Curie proposal, now at 220 pages. Cambridge University Press and the European Mathematical Society have shown serious interest in publishing the book.
A main outcome of the research of Garban, Pete & Schramm on percolation in the plane is that near-critical and dynamical versions of percolation can be very efficiently described using the geometry of the critical system. The methods developed are very robust and can be applied to a wide range of models related to near-critical percolation, such as understanding how exceptionally large clusters in dynamical percolation appear and how they look like (Hammond, Mossel & Pete, and Hammond, Pete & Schramm (see the attached illustration, based on animation by Vincent Beffara); what the continuum planar Minimal Spanning Tree is and what its geometric properties are (Garban, Pete & Schramm); the asymptotic circularity of the supercritical Wulff crystal of percolation (Duminil-Copin); the density of frozen sites in a percolation growth model where clusters freeze when they become large enough (Demeter Kiss (Amsterdam/Cambridge)).
The main results of Duminil-Copin, Garban & Pete are the computation of the near-critical window of the planar FK-Ising model of magnetization via conformal invariance methods and the realization that this window is determined by the critical behaviour of the system in a way very different from percolation --- refuting a natural conjecture made by several leading researchers independently. A possible consequence, work in progress sparked by a conversation with Aizenman (Princeton), is that the famous Harris criterion, predicting when the near-critical behaviour of a disordered system is the same as that of the pure system, might be wrong in many models; if our proposal is correct, it has not been discovered so far because even the way of measuring the effect of disorder has been fundamentally flawed.
Different pieces of work in progress concern the application of noise sensitivity methods developed in earlier work of Garban, Pete & Schramm to a variety of problems: universality results concerning conformal invariance of percolation (with a small result on disordered percolation, noticed first by Aizenman); proving fluctuation bounds in a famous First Passage Percolation problem; exceptional times in dynamical bootstrap percolation on trees (joint with Marek Biskup (UCLA)); the Friedgut-Kalai conjecture on Fourier Entropy versus Influence of general Boolean functions, with a new version suggested by Gábor Tardos (Budapest), possibly applicable to computing the independence ratio of random d-regular graphs (Endre Csóka (Budapest/Warwick)).
Part of the project was (jointly with Elchanan Mossel (Berkeley) and Pablo Shmerkin (Surrey)) to study arithmetic progressions and other patterns in random fractal sets on the real line. Unfortunately, serious new obstacles in the fractal-geometric approach to this problem have been identified. The more combinatorial Szemerédi-type approach is still to be studied; this will soon be undertaken together with József Balogh (Urbana-Champaign), visiting the University of Szeged as a Marie Curie fellow, and Balázs Szegedy (Toronto/Budapest).
GP has been very active in the mathematical life of Hungary. Besides teaching several graduate and undergraduate courses at the Technical University, including a very successful special topics course "Critical phenomena and conformal invariance in the plane", he has been supervising a PhD student at the Central European University (jointly with physicist János Kertész) on the topic of dynamically changing real-world networks, and three MSc and four BSc students at TU Budapest. The MSc research projects have the potential to produce publishable new results. GP has also taken part in restructuring the probability section of the PhD programme of the Central European University, Budapest, and will recurrently teach some graduate courses there. Together with Miklós Abért, he is co-organizing an extremely lively research seminar at the Rényi Institute of the Hungarian Academy of Sciences, entitled "Graphs, groups & probability", attended by many bright young researchers in Hungary. Starting at the end of his MC IIF project, GP has joined the Rényi Institute, and will remain a part-time associate professor at TU Budapest, continuing to teach important probability courses and advise students. New connections between the probability group at TU Budapest and the group theoretical-combinatorial group at the Rényi Institute are shown by a successful Hungarian Scientific Fund (OTKA) grant application, starting Sept 2013, led by Abért, with the participation of GP, Bálint Tóth, and several young researchers from both the Rényi Institute and TU Budapest.
Integration of GP into the Hungarian mathematical community is also shown by his active participation as an OTKA panel member in 2013, by receiving a 3-year Bolyai Fellowship from the Hungarian Academy of Sciences starting Sept 2013, by giving an invited talk at the Erdös Centennial, July 2013, and being a regular speaker at various probability seminars in Budapest. In a broader European context, he has joined the editorial board of the journal Annales de l'Institut Henri Poincaré, and has taught or will teach at several winter schools in Europe (Eurandom 2012, ZiF 2013, Darmstadt 2014).
In summary, the project was very successful both scientifically and regarding transfer of knowledge.