Periodic Reporting for period 4 - NOISE (Noise-Sensitivity Everywhere)
Berichtszeitraum: 2022-08-01 bis 2024-01-31
The study of noise sensitivity originates in computer science, with applications from computational complexity theory to the design of computer chips. One often computes functions of many variables, with a noisy input, but hopes that the noise will not affect the output too much. However, many interesting functions are sensitive even to tiny noise. Perhaps the most striking example comes from statistical physics, in earlier work of the PI with Garban and Schramm: the macroscopic geometry of planar percolation is very sensitive to noise. The aim of this project is to understand what exactly makes a function noise-sensitive, for different types of random input: independent input bits, as typically assumed in theoretical computer science and percolation theory, or different models of statistical mechanics with thermal noise, or random walks in permutation groups. There is a well-established theory for the case of independent random input bits, using discrete Fourier analysis, but almost nothing is known beyond that. The project also intends to apply noise-sensitivity ideas to tackle a variety of unsolved problems at the crossroads of probability theory, statistical physics, group theory, and computer science, such as universality of critical behaviour, existence of random times with exceptional behaviour in different random processes, or the interplay between geometric and probabilistic properties of infinite groups.
The NOISE group, sometimes with coauthors, worked on several aspects of the project, have finished 38 papers, with many manuscripts still being in preparation. We briefly explain six illustrative results of the group.
Tom Hutchcroft (CalTech) and the PI proved a 20-year-old measurable group theory conjecture of Gaboriau, using ideas from statistical physics: infinite Kazhdan groups always have measurable cost equalling to 1. This can be translated to say that there are symmetry-invariant random graphs that make the entire group connected, while having average degree arbitrarily close to 2. So, although Kazhdan groups are big, quickly expanding in one sense, they are almost one-dimensional in another sense. This was the best-known example of a general conjecture, connecting cost and the first ell-2 Betti number (an analytic way to measure "holes") of groups. The paper appeared in Inventiones mathematicae, one of the leading general math journals.
The connection between measurable group theory and statistical physics also works in the other direction: concluding the efforts of several researchers over several decades, Ádám Timár constructed a perfect matching between two independent Poisson point processes, using no extra randomness, with as short connections as possible, using his earlier result on an optimal allocation of volume to a single point process, plus a recent abstract result about measurable graphings.
The PI and Timár disproved the widely believed conjecture that the Free Uniform Spanning Forest in any virtually free (i.e. tree-like) group must be a single tree. They have also provided two different generating sets of the same group such that the FUSF is connected in one Cayley graph, but disconnected in the other. This is probably the first instance discovered when the universality of critical behaviour in a standard statistical physics model does not hold.
Pál Galicza and the PI have established "no sparse reconstruction" results for independent bits and beyond: factor of iid measures and the Ising model, and made connections between sparse reconstruction on the one hand, and cooperative game theory, entropy inequalities, and strong spatial mixing on the other hand. These are first steps in extending noise sensitivity techniques to non-independent inputs.
Alan Hammond (UC Berkeley) and the PI have introduced and analyzed a stochastic game, stake-governed random-tug-of-war. This game is a metaphor for how to optimally convert financial advantage into positional (say, political) advantage. It turned out after the first version of the paper was completed that the game has been studied in the game-theoretic economics literature for decades, without intersections with the mathematics literature. Our paper solves a version that was considered too hard by economists, and the connections to the notorious infinity Laplacian Partial Differential Equation may bring new tools and applications to economics.
The group has obtained many results on the structure of random graphs. One application is to physical networks, which are networks whose nodes and links are not just abstract entities and connections, but physical objects embedded in a geometric space. The PI, with Timár, Stefánsson, Bonamassa, and Pósfai, introduced a new type of dynamically constructed physical network model (a generalization of how the Uniform Spanning Tree of a graph can be built from Loop-Erased Random Walks), together with a new tool (the physical Laplacian) to analyze the effects of physicality on network structure.
Group members have given dozens of seminar and workshop talks all over the world and online. The PI has co-organized summer schools and workshops at the Rényi Institute, and a very lively weekly seminar, hosting many visitors, and also fostering collaboration with other research groups at the Institute. Project member Caio Alves co-created an award-winning YouTube video on Percolation Theory, currently over 344,000 views. At the end of the project, the group organized an international Noise-Sensitivity Workshop with over 30 experts.
Tom Hutchcroft (CalTech) and the PI proved a 20-year-old measurable group theory conjecture of Gaboriau, using ideas from statistical physics: infinite Kazhdan groups always have measurable cost equalling to 1. This can be translated to say that there are symmetry-invariant random graphs that make the entire group connected, while having average degree arbitrarily close to 2. So, although Kazhdan groups are big, quickly expanding in one sense, they are almost one-dimensional in another sense. This was the best-known example of a general conjecture, connecting cost and the first ell-2 Betti number (an analytic way to measure "holes") of groups. The paper appeared in Inventiones mathematicae, one of the leading general math journals.
The connection between measurable group theory and statistical physics also works in the other direction: concluding the efforts of several researchers over several decades, Ádám Timár constructed a perfect matching between two independent Poisson point processes, using no extra randomness, with as short connections as possible, using his earlier result on an optimal allocation of volume to a single point process, plus a recent abstract result about measurable graphings.
The PI and Timár disproved the widely believed conjecture that the Free Uniform Spanning Forest in any virtually free (i.e. tree-like) group must be a single tree. They have also provided two different generating sets of the same group such that the FUSF is connected in one Cayley graph, but disconnected in the other. This is probably the first instance discovered when the universality of critical behaviour in a standard statistical physics model does not hold.
Pál Galicza and the PI have established "no sparse reconstruction" results for independent bits and beyond: factor of iid measures and the Ising model, and made connections between sparse reconstruction on the one hand, and cooperative game theory, entropy inequalities, and strong spatial mixing on the other hand. These are first steps in extending noise sensitivity techniques to non-independent inputs.
Alan Hammond (UC Berkeley) and the PI have introduced and analyzed a stochastic game, stake-governed random-tug-of-war. This game is a metaphor for how to optimally convert financial advantage into positional (say, political) advantage. It turned out after the first version of the paper was completed that the game has been studied in the game-theoretic economics literature for decades, without intersections with the mathematics literature. Our paper solves a version that was considered too hard by economists, and the connections to the notorious infinity Laplacian Partial Differential Equation may bring new tools and applications to economics.
The group has obtained many results on the structure of random graphs. One application is to physical networks, which are networks whose nodes and links are not just abstract entities and connections, but physical objects embedded in a geometric space. The PI, with Timár, Stefánsson, Bonamassa, and Pósfai, introduced a new type of dynamically constructed physical network model (a generalization of how the Uniform Spanning Tree of a graph can be built from Loop-Erased Random Walks), together with a new tool (the physical Laplacian) to analyze the effects of physicality on network structure.
Group members have given dozens of seminar and workshop talks all over the world and online. The PI has co-organized summer schools and workshops at the Rényi Institute, and a very lively weekly seminar, hosting many visitors, and also fostering collaboration with other research groups at the Institute. Project member Caio Alves co-created an award-winning YouTube video on Percolation Theory, currently over 344,000 views. At the end of the project, the group organized an international Noise-Sensitivity Workshop with over 30 experts.
All six results mentioned in the previous section are examples of unexpected developments that go well beyond the state of the art and may have a lasting impact in some areas of probability theory, statistical physics, group theory, game theory, and network science, especially regarding the interplay of these areas with each other.
On a longer run, results from the project might challenge the apparently naive believe, widespread in science and technology, that important functions are noise stable. (For instance, Gil Kalai, one of the founders of the theory of noise sensitivity, speculates that noise sensitivity might make effective quantum computing impossible, and it furthermore might be an underlying reason for the existence of dark matter and energy.)
On a longer run, results from the project might challenge the apparently naive believe, widespread in science and technology, that important functions are noise stable. (For instance, Gil Kalai, one of the founders of the theory of noise sensitivity, speculates that noise sensitivity might make effective quantum computing impossible, and it furthermore might be an underlying reason for the existence of dark matter and energy.)