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ERC Consolidator Grant 772466
NoiseSensitivity Everywhere (NOISE)
Gábor Pete
Alfréd Rényi Institute of Mathematics
February 2018  January 2024
Press release of highlighted projects by the European Research Council, including mine.
Postdoc position open now.
Website at the Rényi Institute.
Mathematical summary:
Noisesensitivity of a Boolean function with iid random input bits means that resampling a tiny proportion of the input makes the output unpredictable. This notion arises naturally in computer science, but perhaps the most striking example comes from statistical physics, in large part due to the PI: the macroscopic geometry of planar percolation is very sensitive to noise. This can be recast in terms of Fourier analysis on the hypercube: a function is noise sensitive iff most of its Fourier weight is on “high energy” eigenfunctions of the random walk operator.
This project proposes to use noise sensitivity ideas in three main directions:

Address some outstanding questions in the classical case of iid inputs: universality in critical planar percolation; the FriedgutKalai conjecture on Fourier Entropy vs Influence; noise in First Passage Percolation.

In statistical physics, a key example is the critical planar FKIsing model, with noise being Glauber dynamics. One task is to prove noise sensitivity of the macroscopic structure. A key obstacle is that hypercontractivity of the critical dynamics is not known.

Babai's conjecture says that random walk on any finite simple group, with any generating set, mixes
in time polylogarithmic in the volume. Two key open cases are the alternating groups and the linear groups SL(n, F2). We will approach these questions by first proving fast mixing for certain macroscopic structures. For permutation groups, this is the cycle structure, and it is related to a conjecture of Balint Toth on the interchange process, motivated by a phase transition question in quantum mechanics.
We will also apply ideas of statistical physics to group theory in other novel ways, such as understanding the relation between the first ell2Betti number of a group and its measurable cost, or using random walks in random environment to prove amenability of certain groups.
Project members (present and past):
Ádám Timár (Senior researcher 2019 Rényi)
Percolation processes and unimodular random graphs.
Péter Mester (Parttime senior researcher 2020 Rényi)
Groupinvariant percolation processes.
Miklós Abért (Senior researcher, parttime advising the project, 2023 Rényi)
Geometric group theory, measurable group actions, unimodularity beyond graphs.
Bálint Tóth (Senior researcher, parttime advising the project, 2023 Rényi)
Random walks in random environments. The interchange process and the quantum Heisenberg model.
Balázs Ráth (Senior researcher, parttime advising the project, 2023 Rényi)
Percolation processes, selforganized criticality, endogeny, random interlacements.
Adam Arras (PostDoc 2023 Rényi)
Spectral theory of random graphs.
Zsolt Bartha (PostDoc 2023 Rényi)
Bootstrap percolation, constrained satisfaction problems on random graphs.
Jacob Richey (PostDoc 2023 Rényi)
All sorts of discrete probability: interacting particle systems, subshifts of finite type, random graphs.
András Tóbiás (PostDoc 2023 Rényi)
Geometric percolation and population genetics processes.
László Márton Tóth (PostDoc 2023 Rényi)
Measurable group actions, factor of iid processes.
Caio Alves (PostDoc 20202022 Rényi, currently at Oak Ridge National Lab)
Percolation theory, loop soup, random graphs.
Olle Elias (PostDoc 20202022 Rényi, currently at Uni Köln)
Percolation theory, interlacements.
Ábel Farkas (PostDoc 20182020 Rényi, currently doing improv theatre)
Fractal percolation, geometric measure theory.
Pál Galicza (PhD student 20142020 CEU, PostDoc 20202023 Rényi)
Noise sensitivity of Boolean functions and percolation. Sparse reconstruction in spin systems.
Ágnes Cs. Kúsz (PhD student 2020 BME)
Random trees.
Sándor Rokob (PhD student 2018 BME, coadvised with Balázs Ráth)
Random interlacements and other Poissonian percolation models, Uniform Spanning Forests.
Márton Szőke (PhD student 2021 BME, advised by Balázs Ráth)
Stochastic processes on random graphs. Endogeny questions.
Richárd Patkó (PhD student 20172018 BME, currently at Bosch)
Representation theory and random walks on groups
Mahefa Ravelonanosy (MSc student 2020 CEU, Research intern 2020 Rényi, currently PhD student at TU Eindhoven)
Concentration of distances in graph sequences
Gergő Lukáts (MSc student 2019 BME, currently PhD student at Univ Oslo)
Mixing time of critical Ising Glauber dynamics
Some papers and talks:

Zsolt Bartha, Júlia Komjáthy, Daniel Valesin.
Degreepenalized contact processes. Preprint, 71 pages. [2310.07040 math.PR]

János Engländer, Giulio Iacobelli, Gábor Pete, Rodrigo Ribeiro.
Structural results for the Tree Builder Random Walk. Preprint, 34 pages. [arXiv:2311.18734 math.PR] Here is a talk at the Oxford Discrete Maths and Probability Seminar.

Gábor Elek and Ádám Timár.
A comprehensive characterization of Property A and almost finiteness. Preprint, 44 pages. [arXiv:2308.14554 math.GR]

Ádám Timár.
Factor of iid's through stochastic domination. Preprint, 11 pages. [arXiv:2306.15120 math.PR]

Gábor Pete, Ádám Timár, Sigurdur Örn Stefánsson, Ivan Bonamassa, Márton Pósfai.
A networkofnetworks model for physical networks. Preprint, 8 pages. [arXiv:2306.01583 condmat.statmech] Here is a talk at the Oxford Discrete Maths and Probability Seminar.

Márton Borbényi, Balázs Ráth, Sándor Rokob.
Random interlacement is a factor of i.i.d. Elect. J. Probab. 28 (2023), 145. [arXiv:2208.14545 math.PR]

Alan Hammond and Gábor Pete,
Stakegoverned tugofwar and the biased infinity Laplacian. Preprint, 69 pages. [arXiv:2206.08300 math.PR]

Alexander Drewitz, Olof Elias, Alexis Prévost, Johan Tykesson, and Fredrik Viklund.
Percolation for twodimensional excursion clouds and the discrete Gaussian free field. Preprint, 58 pages. [arXiv:2205.15289 math.PR]

Caio Alves and Augusto Teixeira.
Cylinders' percolation: decoupling and applications. Preprint, 42 pages. [arXiv:2112.10055 math.PR] Here is their talk at Percolation Today.

Balázs Ráth and Sándor Rokob.
Percolation of worms. Stoch. Proc. Appl. 152 (2022), 233288. [arXiv:2107.03259 math.PR] Here is their talk at ISI Bangalore.

Ádám Timár.
A factor matching of optimal tail between Poisson processes. Combinatorica (2023), 5 pages. [arXiv:2106.04524 math.PR]

Caio Alves, Rangel Baldasso, Gideon Amir, and Augusto Teixeira.
Percolation phase transition on planar spin systems. Preprint, 40 pages. [arXiv:2105.13314 math.PR]

Balázs Ráth, Jan M. Swart, Márton Szőke.
A phase transition between endogeny and nonendogeny. Elect. J. Probab. (2022), 44 pages. [arXiv:2103.14408 math.PR]

Pál Galicza and Gábor Pete.
Sparse reconstruction in spin systems I: iid spins. Israel J Math, to appear, 54 pages. [arXiv:2010.10483 math.PR] Here are some talk slides on the project, Prague Stochastics, August 2019.

Gábor Pete and Ádám Timár.
The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set. Ann. Probab. 50 (2022) 22182243. [arXiv:2006.06387 math.PR] Here is our talk at Percolation Today.

Tom Hutchcroft and Gábor Pete.
Kazhdan groups have cost 1. Inventiones mathematicae, vol. 221 (2020), pages 873891. [arXiv:1810.11015 math.GR] Here is my talk at IIAS, Jerusalem, October 2018.

Itai Benjamini and Ádám Timár.
Invariant embeddings of unimodular random planar graphs. Elect. J. Probab. (2021), 24 pages. [arXiv:1910.01614 math.PR]

Pál Galicza.
Pivotality versus noise stability for monotone transitive functions. Elect. Comm. Probab., Volume 25 (2020), paper no. 17, 6 pp. [arXiv:1909.05375 math.PR]

Richárd Patkó and Gábor Pete.
Mixing time and cutoff phenomenon for the interchange process on dumbbell graphs and the labelled exclusion process on the complete graph. Preprint, 21 pages. [arXiv:1908.09406 math.PR]

Ábel Farkas.
Conditional measure on the Brownian path and other random sets. Preprint, 89 pages. [arXiv:1704.05745 math.PR]

Péter Mester.
A factor of i.i.d with uniform marginals and infinite clusters spanned by equal labels. Ergodic Th. Dyn. Systems, to appear, 9 pages. [arXiv:1111.3067v2 math.PR]
An awardwinning visual introduction to percolation theory by the Spectral Collective (Caio Alves, Aranka Hrušková, Vilas Winstein), Summer of Math Exposition, 2022.
(Last updated: February 2023.)