Random Walks and Quantum Spin Systems
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
Random motions with long memory:
Markovian random walks (with short memory) in d-dimensional space have been studied since the early years of the twentieth century as simple models of various phenomena (like diffusion of particles suspended in fluids in thermal equilibrium). By now all aspects of these random processes are fully understood. However, true physical (and other natural) phenomena are much more complicated than these naïve models and the methods of classical probability simply do not suffice for their study. In particular, due to interactions of the observed moving particle with its environment long-time correlations build up and therefore the naïve Markovian approximations become unsuitable. Since the early 1980s very intense research has been concentrated on understanding more realistic mathematical models of diffusion-like phenomena. Among the most investigated classes of models are the following:
-- Random walks and diffusions in random environment, where the local rules of the random walker are spatially inhomogeneous and randomly sampled themselves.
-- Random walks with self-interactions, where the random walker's local rules are influenced by its own past trajectory via some local functional of its own occupation time measure.
-- Diffusion under deterministic (typically Hamiltonian) dynamics, where the randomness comes only with the initial conditions of the system, e.g. due to thermal equilibrium.
Our research ambition is to understand the long-time asymptotic scaling behaviour of these processes. We will study the long-time asymptotics (so-called scaling limits) of these processes, proving normal or anomalous diffusion in relevant models.
Stochastic representations for quantum spin systems:
More than eighty years since its formulation the quantum Heisenberg model of interacting spins is still a fundamental model of quantum statistical physics. It is sufficiently rich to encode complex physical phenomena and pose deep mathematical challenges. The main problem is the existence of so-called off-diagonal long range order - a kind of magnetic ordering - in particular instances. The relevance of the problem is emphasized by noting that off-diagonal long range order is equivalent to Bose-Einstein condensation, thus being of paramount importance in understanding of superfluidity. In a ground breaking work published in 1978 Dyson, Lieb and Simon (DLS) proved the occurrence of off-diagonal long-range order at low positive temperatures in 3 dimensions, for models with antiferromagnetic interactions. Since then a similar result for ferromagnetic couplings escapes all attempts of rigorous mathematical proof. The main problem here is establishing a particular correlation inequality called infrared bound. The method of DLS substantially relies on a particular feature of the antiferromagnetic models, called reflection positivity, which simply doesn't hold in ferromagnetic cases. Nevertheless, the infrared bound is expected to hold. A major challenge for the specialists working in this field is to find some way around the reflection positivity argument and arrive at the truly relevant correlation inequalities (infrared bounds) by other means.
Stochastic representations of the quantum spin systems arise via a beautiful link with probability, the Feynman-Kac formula, and recently became rather popular, since they lead to probabilistic reformulations of the relevant quantum statistical physics problems, typically in terms of interacting stochastic particle systems or stochastic geometric objects, like random loops on graphs.
My main ambition in this context is to apply a well suited probabilistic reformulation of the spin-1/2 isotropic quantum Heisenberg model with ferromagnetic couplings, reformulate the infrared bound as a probabilistic correlation inequality for a particular stochastic interacting particle system (the so-called symmetric simple exclusion model) and prove it this way.
Markovian random walks (with short memory) in d-dimensional space have been studied since the early years of the twentieth century as simple models of various phenomena (like diffusion of particles suspended in fluids in thermal equilibrium). By now all aspects of these random processes are fully understood. However, true physical (and other natural) phenomena are much more complicated than these naïve models and the methods of classical probability simply do not suffice for their study. In particular, due to interactions of the observed moving particle with its environment long-time correlations build up and therefore the naïve Markovian approximations become unsuitable. Since the early 1980s very intense research has been concentrated on understanding more realistic mathematical models of diffusion-like phenomena. Among the most investigated classes of models are the following:
-- Random walks and diffusions in random environment, where the local rules of the random walker are spatially inhomogeneous and randomly sampled themselves.
-- Random walks with self-interactions, where the random walker's local rules are influenced by its own past trajectory via some local functional of its own occupation time measure.
-- Diffusion under deterministic (typically Hamiltonian) dynamics, where the randomness comes only with the initial conditions of the system, e.g. due to thermal equilibrium.
Our research ambition is to understand the long-time asymptotic scaling behaviour of these processes. We will study the long-time asymptotics (so-called scaling limits) of these processes, proving normal or anomalous diffusion in relevant models.
Stochastic representations for quantum spin systems:
More than eighty years since its formulation the quantum Heisenberg model of interacting spins is still a fundamental model of quantum statistical physics. It is sufficiently rich to encode complex physical phenomena and pose deep mathematical challenges. The main problem is the existence of so-called off-diagonal long range order - a kind of magnetic ordering - in particular instances. The relevance of the problem is emphasized by noting that off-diagonal long range order is equivalent to Bose-Einstein condensation, thus being of paramount importance in understanding of superfluidity. In a ground breaking work published in 1978 Dyson, Lieb and Simon (DLS) proved the occurrence of off-diagonal long-range order at low positive temperatures in 3 dimensions, for models with antiferromagnetic interactions. Since then a similar result for ferromagnetic couplings escapes all attempts of rigorous mathematical proof. The main problem here is establishing a particular correlation inequality called infrared bound. The method of DLS substantially relies on a particular feature of the antiferromagnetic models, called reflection positivity, which simply doesn't hold in ferromagnetic cases. Nevertheless, the infrared bound is expected to hold. A major challenge for the specialists working in this field is to find some way around the reflection positivity argument and arrive at the truly relevant correlation inequalities (infrared bounds) by other means.
Stochastic representations of the quantum spin systems arise via a beautiful link with probability, the Feynman-Kac formula, and recently became rather popular, since they lead to probabilistic reformulations of the relevant quantum statistical physics problems, typically in terms of interacting stochastic particle systems or stochastic geometric objects, like random loops on graphs.
My main ambition in this context is to apply a well suited probabilistic reformulation of the spin-1/2 isotropic quantum Heisenberg model with ferromagnetic couplings, reformulate the infrared bound as a probabilistic correlation inequality for a particular stochastic interacting particle system (the so-called symmetric simple exclusion model) and prove it this way.
Planned Impact
Impact within our scientific community:
The problems proposed for investigation are in the main stream of current research in probability theory and its applications. There are regular, very prestigious research programmes and conferences dedicated to these and closely related subjects. Recent examples include The Mittag Leffler Institute Stockholm (Spring 2009), The Fields Institute Toronto (Spring 2011), MSRI Berkeley (Spring 2012), the Mathematics Institute of Warwick University (Spring, 2014), the Isaac Newton Institute Cambridge (Spring 2015), where I had been invited. In Spring 2017 I will participate in a similar program at Institut Henri Poincaré, Paris. I expect similar invitations for the future years, too. The PDRA will also take part and present his research in contributed sessions at similar events. This is clear evidence that any progress made will have a significant impact on both national and international top research. We expect to obtain mathematical results of high scientific value and appreciated by the mathematical community and these will have influence within the mathematics community working on various aspects of probability theory and stochastic processes, with particular emphasis on walks and diffusions in inhomogeneous and random environment. Our results will also have impact on theoretical and mathematical physicists working in statistical and condensed matter physics, on biologists of mathematical orientation working on population dynamics and on network scientists.
Dissemination of our results will be achieved primarily via the traditional venues of mathematical sciences: publications in top rank journals, talks and presentations at prestigious international conferences, active participation in relevant workshops, survey papers and colloquia for wider audiences. As demonstrated by my track record, I publish my research results in leading probability, and mathematical physics journals. All my papers are freely accessible at the academic preprint repository arXiv. These are the main channels of dissemination in our profession.
I also plan to give further postgraduate courses at summer schools - as I have done regularly in the past. These will attract prospective young researchers to these topics.
I am involved in organisation of a number of conferences and workshops on these themes in the coming years. During the first half of this fellowship I will organise three international conferences (Oberwolfach DE, 2016 Nov; Luminy-Marseille FR, 2017 May; Bristol UK, 2018 Mar; all sponsored from other sources than EPSRC) and two workshops (both in Bristol, sponsored by the Leverhulme Trust) in the topics of this proposal. These events will also provide excellent opportunities for dissemination of our research achievements.
Impact outside our scientific community:
I have a wide teaching record across all levels of university teaching, and this has hugely benefited from active mathematical research. Continued support of our research activities thus benefits a large group of prospective and current students in higher education.
As public engagement:
The University of Bristol is in the process of launching a bimonthly Big Science public talk series which is coordinated by the Public Engagement Team and champions the outstanding research stories in the Faculty of Science. I will also give a public talk about his research in this flagship series of public talks. Beside this, I undertook to present a talk in the very popular and prestigious series Bristol Public Talks in Mathematics.
The problems proposed for investigation are in the main stream of current research in probability theory and its applications. There are regular, very prestigious research programmes and conferences dedicated to these and closely related subjects. Recent examples include The Mittag Leffler Institute Stockholm (Spring 2009), The Fields Institute Toronto (Spring 2011), MSRI Berkeley (Spring 2012), the Mathematics Institute of Warwick University (Spring, 2014), the Isaac Newton Institute Cambridge (Spring 2015), where I had been invited. In Spring 2017 I will participate in a similar program at Institut Henri Poincaré, Paris. I expect similar invitations for the future years, too. The PDRA will also take part and present his research in contributed sessions at similar events. This is clear evidence that any progress made will have a significant impact on both national and international top research. We expect to obtain mathematical results of high scientific value and appreciated by the mathematical community and these will have influence within the mathematics community working on various aspects of probability theory and stochastic processes, with particular emphasis on walks and diffusions in inhomogeneous and random environment. Our results will also have impact on theoretical and mathematical physicists working in statistical and condensed matter physics, on biologists of mathematical orientation working on population dynamics and on network scientists.
Dissemination of our results will be achieved primarily via the traditional venues of mathematical sciences: publications in top rank journals, talks and presentations at prestigious international conferences, active participation in relevant workshops, survey papers and colloquia for wider audiences. As demonstrated by my track record, I publish my research results in leading probability, and mathematical physics journals. All my papers are freely accessible at the academic preprint repository arXiv. These are the main channels of dissemination in our profession.
I also plan to give further postgraduate courses at summer schools - as I have done regularly in the past. These will attract prospective young researchers to these topics.
I am involved in organisation of a number of conferences and workshops on these themes in the coming years. During the first half of this fellowship I will organise three international conferences (Oberwolfach DE, 2016 Nov; Luminy-Marseille FR, 2017 May; Bristol UK, 2018 Mar; all sponsored from other sources than EPSRC) and two workshops (both in Bristol, sponsored by the Leverhulme Trust) in the topics of this proposal. These events will also provide excellent opportunities for dissemination of our research achievements.
Impact outside our scientific community:
I have a wide teaching record across all levels of university teaching, and this has hugely benefited from active mathematical research. Continued support of our research activities thus benefits a large group of prospective and current students in higher education.
As public engagement:
The University of Bristol is in the process of launching a bimonthly Big Science public talk series which is coordinated by the Public Engagement Team and champions the outstanding research stories in the Faculty of Science. I will also give a public talk about his research in this flagship series of public talks. Beside this, I undertook to present a talk in the very popular and prestigious series Bristol Public Talks in Mathematics.
Organisations
- University of Bristol (Fellow, Lead Research Organisation)
- Georgia Institute of Technology (Collaboration)
- Technion - Israel Institute of Technology (Collaboration)
- University of Wisconsin-Madison (Collaboration)
- UNIVERSITY OF CAMBRIDGE (Collaboration)
- University of California, Berkeley (Collaboration)
- University of Melbourne (Collaboration)
- University of Amsterdam (Collaboration)
- Rutgers, The State University of New Jersey (Collaboration)
- Weizmann Institute of Science (Collaboration)
- Roma Tre University (Collaboration)
- UNIVERSITY OF BRITISH COLUMBIA (Collaboration)
- University of Illinois at Chicago (Collaboration)
- University of Bristol (Collaboration)
People |
ORCID iD |
| Balint Toth (Principal Investigator / Fellow) |
Publications
Bauerschmidt R
(2020)
Random Spanning Forests and Hyperbolic Symmetry
in Communications in Mathematical Physics
Bauerschmidt R
(2019)
Dynkin isomorphism and Mermin-Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process
in Annals of Probability
Bodineau T
(2020)
Large Scale Stochastic Dynamics
in Oberwolfach Reports
Bodineau T
(2017)
Large Scale Stochastic Dynamics
in Oberwolfach Reports
Borgs C
(2022)
Efficient sampling and counting algorithms for the Potts model on Z d at all temperatures
in Random Structures & Algorithms
Brydges D. C.
(2021)
The continuous-time lace expansion
in Communications in Pure and Applied Mathematics
Crane E
(2019)
Diffusion and Superdiffusion in Lattice Models for Colliding Particles with Stored Momentum
in Journal of Statistical Physics
Crane E.
(2019)
Diffusion and superdiffusion in lattice models for colliding particles with stored momentum
in Journal of Statistical Physics
Hammond A
(2019)
Self-attracting self-avoiding walk
in Probability Theory and Related Fields
| Description | /////////////////////////////////////////// (A) RANDOM PROCESSES WITH LONG MEMORY, RANDOM WALKS IN RANDOM ENVIRONMENTS: ******* (A1) Proved annealed and quenched central limit theorems for random walks in divergence-free random drift field, under arguably optimal conditions (the so-called H_{-1} condition) on the correlations of the drift field. These results are of fundamental importance in the context of random walks and diffusions in disordered media. Two papers published in The Annals of Probability, leading journal of probability theory. [https://projecteuclid.org/euclid.aop/1513069261, https://projecteuclid.org/euclid.aop/1537862440] ******* (A2) Proved the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range step distribution. The proof relies on a direct connection between the VRJP and spin models whose target space is a hyperbolic space or its supersymmetric counterpart and a new version of the celebrated Mermin-Wagner theorem for spin-spaces with a non-compact symmetry group action. Two papers published/accepted for publication in leading journal of the field, The Annals of Probability, resp., Annales Henri Poincaré. [https://projecteuclid.org/euclid.aop/1571731454, https://arxiv.org/abs/1904.01532] ******* (A3) Scaling limit proved for random walks in high dimensions, where self-attractive and self-repelling forces compete. The result is a tour-de-force of mathematically rigorously controlled perturbative methods , the so-called lace expansion. Published in Probability Theory and Related Fields, leading journal of the field. [https://link.springer.com/article/10.1007/s00440-018-00898-7] ******* (A4) Proved (anomalous) superdiffusive bounds for random walks in divergence-free drift field in some cases when the H_{-1} condition fails to hold. A case of particular interest is the so-called randomly oriented Manhattan-lattice (originating in physics literature). We apply the resolvent method developed in earlier work. The problem has some notoriety, so the result is appreciated by the probability community. Published in a strong journal of the field. [https://projecteuclid.org/euclid.ecp/1532505674] /////////////////////////////////////////// (B) DIFFUSION FROM DETERMINISTIC DYNAMICS WITH RANDOM INITIAL CONDITIONS: ******* (B1) Proved invariance principle (a.k.a. functional central limit theorem) for the classical Lorentz gas and for the Ehrenfest wind-tree model, with randomly placed scatterers, in a scaling regime going beyond the so-called Boltzmann-Grad (or low density) limit. These results represent significant progress in a major problem of nonequilibrium statistical physics. Two papers published/accepted for publication in leading journals of the field, Communications in Mathematical Physics, resp., Annales Henri Poincaré [https://link.springer.com/article/10.1007%2Fs00220-020-03852-8, https://arxiv.org/abs/1912.02492] /////////////////////////////////////////// (C) STOCHASTIC REPRESENTATION OF CLASSICAL AND QUANTUM SPIN SYSTEMS AND LATTICE GASES: ******* (C1) Proved subtle result about geometric properties of random permutation processes on large d-dimentional discrete tori. The investigation is motivated by lack of mathematically satisfactory understanding of phase transitions and emergence of long-range ordering in quantum spin systems and bosonic quantum lattice gases (a.k.a. Bose-Einstein condensation). Published in a strong journal of mathematical physiscs. [https://link.springer.com/article/10.1007/s10955-020-02487-2] ******* (C2) Reformulate an existing result (of Caracciolo et al), showing that Bernoulli bond percolation conditioned on being a forest can be rephrased as a fermionic spin system with hyperbolic symmetry. The reformulation allows using dimensional reduction and horospherical coordinates to obtain explicit integral formulas for connection probabilities in terms of a real-valued spin system. This formula enables to implement a Mermin-Wagner theorem to show the model does not exhibit a phase transition two-dimensions, in contrast with the same model's behaviour in higher dimension. Published in Communications in Mathematical Physics, leading journal of the field. [https://link.springer.com/article/10.1007/s00220-020-03921-y] ******* (C3) Gave mathematically rigorous version of a heuristic/phenomenological observation of physicists that loop-erased random walk should be related to a phi^4 field theory. The formulation shows that a field theory with 2 complex fermions and 1 complex boson encodes loop-erased random walks, and correlation functions of this field theory can be used to measure statistical properties of loop-erased random walk like the one-point function. Published in a strong journal of mathematical physiscs. [https://link.springer.com/article/10.1007%2Fs10955-020-02628-7] ******* (C4) See also (D2) below. /////////////////////////////////////////// (D) DISCRETE STATISTICAL PHYSICS, COMPUTER SCIENCE, ALGORITHMS: ******* (D1) Developed efficient algorithmic methods for low-temperature counting problems in statistical physics. Gave efficient approximate counting and sampling algorithms for the Potts model on Z^d, with fixed high value of colours, at ALL temperatures. This result rigorously rules out the existence of any computational phase transition associated to the first-order physical phase transition of the model on subgraphs of Z^d. The methods provide a blueprint for using Pirogov-Sinai theory in the presence of unstable ground states. Presented and publishe in the proceedings of two consecutive editions of the most prominent international annual event in theoretical computer science, the 51st and 52nd ACM Symposium on Theory of Computing (STOC-2019, STOC-2020) [https://dl.acm.org/doi/10.1145/3313276.3316305, https://dl.acm.org/doi/10.1145/3357713.3384271]. Expanded version with complete mathematical background and proof published in Probability Theory and Related Fields, leading journal of the field. [https://link.springer.com/article/10.1007/s00440-019-00928-y] ******* (D2) Found optimal route to proving some recent relevant results concerning the approximate computation of high-temperature quantum partition functions by classical algorithms. The method suggests that similar results may be possible for some quantum spin systems at low temperatures. Published in leading journal of mathematical physics. [Tyler Helmuth (PDRA) with coauthor, https://aip.scitation.org/doi/10.1063/5.0013689] /////////////////////////////////////////// |
| Exploitation Route | The mathematical results become organic parts of mainstream modern probability theory and mathematically rigorous statistical physics. The technical tools and methodology developed and employed are likely to be used in various other contexts. |
| Sectors | Education,Other |
| Description | Standard individual research grant |
| Amount | 6,504,000 Ft (HUF) |
| Organisation | National Research, Development and Innovation Office (Hungary) |
| Sector | Public |
| Country | Hungary |
| Start | 12/2018 |
| End | 11/2022 |
| Description | Support of organizing workshop on "High-dimensional critical phenomena in random environments" Bristol, Sep. 24--26, 2018 |
| Amount | £7,000 (GBP) |
| Organisation | Heilbronn Institute for Mathematical Research |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 09/2018 |
| End | 09/2018 |
| Description | Supporting the organisation of international conference "Probability, Analysis and Dynamics" Bristol, 5-6 April 2018 |
| Amount | £20,000 (GBP) |
| Organisation | Heilbronn Institute for Mathematical Research |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 04/2018 |
| End | 06/2018 |
| Description | Travel support for ICM-2018 |
| Amount | £1,500 (GBP) |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 08/2018 |
| End | 09/2018 |
| Description | Georgia Institute of Technology, USA - College of Computing - research collaboration |
| Organisation | Georgia Institute of Technology |
| Department | College of Computing |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | C. Borgs, J. Chayes, T. Helmuth, W. Perkins, P. Tetali: Efficient sampling and counting algorithms for the Potts model on Z^d at all temperatures, STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, June 2020 Pages 738-751 [https://arxiv.org/abs/1909.09298] [Contributor from Georgia Institute of Technology: P. Tetali] |
| Start Year | 2017 |
| Description | Rutgers University, USA - Department of Mathematics - research collaboration |
| Organisation | Rutgers University |
| Department | Department of Mathematics |
| Country | United States |
| Sector | Private |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | C. Lutsko, B. Toth: Invariance Principle for the Random Lorentz Gas - Beyond the Boltzmann-Grad Limit. Communiations in Mathematical Physics 379, 589-632 (2020) [Contributor from Rutgers University: C. Lutsko] C. Lutsko, B. Toth: Invariance Principle for the Random Wind-Tree Process. Annales Henri Poincaré (2021, to appear) [Contributor from Rutgers University: C. Lutsko] |
| Start Year | 2020 |
| Description | Technion, Israel Institute of Technology, IL - Departments of Mathematics and of Industrial Engineering- research collaboration |
| Organisation | Technion - Israel Institute of Technology |
| Department | Department of Mathematics |
| Country | Israel |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | R. Bauerschmidt, N. Crawford, T. Helmuth, A. Swan: Random spanning forests and hyperbolic symmetry, Commun. Math. Phys. (2021, to appear) [https://arxiv.org/abs/1912.04854] [Collaborator from Technion: N. Crawford] D. Ioffe, B. Toth: Split-and-Merge in Stationary Random Stirring on Lattice Torus. Journal of Statistical Physics 180: 630-653 (2020) [arXiv:1909.06188] [Collaborator from Technion: D. Ioffe] |
| Start Year | 2016 |
| Description | Technion, Israel Institute of Technology, IL - Departments of Mathematics and of Industrial Engineering- research collaboration |
| Organisation | Technion - Israel Institute of Technology |
| Department | Faculty of Industrial Engineering and Management |
| Country | Israel |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | R. Bauerschmidt, N. Crawford, T. Helmuth, A. Swan: Random spanning forests and hyperbolic symmetry, Commun. Math. Phys. (2021, to appear) [https://arxiv.org/abs/1912.04854] [Collaborator from Technion: N. Crawford] D. Ioffe, B. Toth: Split-and-Merge in Stationary Random Stirring on Lattice Torus. Journal of Statistical Physics 180: 630-653 (2020) [arXiv:1909.06188] [Collaborator from Technion: D. Ioffe] |
| Start Year | 2016 |
| Description | The Weizmann Institute of Science, Israel - Department of Mathematics - research collaboration |
| Organisation | Weizmann Institute of Science |
| Department | Department of Mathematics |
| Country | Israel |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | G. Kozma, B. Toth: Central limit theorem for random walks in divergence-free random drift field: H_{-1} suffices. The Annals of Probability, 45 (2017) 4307-4347 [arXiv:1702.06905] [Contributor from the Weizmann Institute: G. Kozma] |
| Start Year | 2016 |
| Description | University Roma Tre, IT - Department of Mathematics and Physics - research collaboration |
| Organisation | Roma Tre University |
| Department | Department of Mathematics and Physics |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | T. Helmuth, A. Shapira: Loop-erased random walk as a spin system observable, J. Stat. Phys. 181(4), 1306-1322, 2020. [https://arxiv.org/abs/2003.10928] [Collaborator from University Roma Tre: A. Shapira] |
| Start Year | 2019 |
| Description | University of Amsterdam, NL - Korteweg de Vries Institute for Mathematics - research collaboration |
| Organisation | University of Amsterdam |
| Country | Netherlands |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | T. Helmuth, W. Perkins, G. Regts: Algorithmic Pirogov-Sinai theory, Probab. Theory and Relat. Fields, 176, 851-895, (2020) [https://arxiv.org/abs/1806.11548] [Contributor from the University of Amsterdam: G. Regts] T. Helmuth, W. Perkins, G. Regts: Algorithmic Pirogov-Sinai theory (STOC-2019), STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of ComputingJune 2019 Pages 1009-1020, [https://arxiv.org/abs/1806.11548] [Contributor from the University of Amsterdam: G. Regts] |
| Start Year | 2017 |
| Description | University of Bristol, UK - School of Mathematics - research collaboration |
| Organisation | University of Bristol |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | T. Helmuth, R.L. Mann: Efficient algorithms for approximating quantum partition functions, J. Math. Phys.(2021, to appear) [https://arxiv.org/abs/2004.11568] [Collaborator from the University of Bristol (not member of the EPSRC supported team): R.L. Mann] C. Lutsko, B. Toth: Invariance Principle for the Random Lorentz Gas - Beyond the Boltzmann-Grad Limit, Communiations in Mathematical Physics 379, 589-632 (2020) [arXiv:1812.11325] [Collaborator from the University of Bristol (not member of the EPSRC supported team): C. Lutsko] C. Lutsko, B. Toth: Invariance Principle for the Random Wind-Tree Process, Annales Henri Poincaré (2021, to appear) [arXiv:1912.02492] [Collaborator from the University of Bristol (not member of the EPSRC supported team): C. Lutsko] E. Crane, S. Ledger, B. Toth: Diffusion and superdiffusion in lattice models for colliding particles with stored momentum, Journal of Statistical Physics 177: 1240-1262 (2019) [arXiv:1809.03257] [Collaborators from the University of Bristol (not member of the EPSRC supported team): E. Crane, S. Ledger] S. Ledger, B. Toth, B. Valko: Random walk on the randomly oriented Manhattan lattice. Electronic Communications in Probability 23, paper no. 43 (2018) [arXiv:1802.01558] [Collaborator from the University of Bristol (not member of the EPSRC supported team): S. Ledger] |
| Start Year | 2016 |
| Description | University of British Columbia, Canada - Department of Mathematics - research collaboration |
| Organisation | University of British Columbia |
| Department | Department of Mathematics |
| Country | Canada |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | D. C. Brydges, T. Helmuth, M. Holmes, "The continuous-time lace expansion", Commun. Pure Appl. Math, (2021, to appear) [https://arxiv.org/abs/1905.09605] [Contributor from the University of British Columbia: D. C. Brydges] |
| Start Year | 2017 |
| Description | University of California, Berkeley, USA - Departments of Mathematics and Statistics - research collaboration |
| Organisation | University of California, Berkeley |
| Department | Department of Mathematics |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | C. Borgs, J. Chayes, T. Helmuth, W. Perkins, P. Tetali: Efficient sampling and counting algorithms for the Potts model on Z^d at all temperatures, STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, June 2020 Pages 738-751 [https://arxiv.org/abs/1909.09298] [Contributors from the University of California at Berkeley: C. Borgs, J. Chayes] A. Hammond, T. Helmuth: Self-attracting self-avoiding walk, Probab. Theory and Relat. Fields, 175(3-4), 677-719 (2019) [https://arxiv.org/abs/1712.07673] [Contributor from the University of California at Berkeley: A. Hammond] |
| Start Year | 2017 |
| Description | University of California, Berkeley, USA - Departments of Mathematics and Statistics - research collaboration |
| Organisation | University of California, Berkeley |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | C. Borgs, J. Chayes, T. Helmuth, W. Perkins, P. Tetali: Efficient sampling and counting algorithms for the Potts model on Z^d at all temperatures, STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, June 2020 Pages 738-751 [https://arxiv.org/abs/1909.09298] [Contributors from the University of California at Berkeley: C. Borgs, J. Chayes] A. Hammond, T. Helmuth: Self-attracting self-avoiding walk, Probab. Theory and Relat. Fields, 175(3-4), 677-719 (2019) [https://arxiv.org/abs/1712.07673] [Contributor from the University of California at Berkeley: A. Hammond] |
| Start Year | 2017 |
| Description | University of Cambridge, UK - Department of Pure Mathematics and Mathematical Statistics - research collaboration |
| Organisation | University of Cambridge |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | R. Bauerschmidt, N. Crawford, T. Helmuth, A. Swan: Random spanning forests and hyperbolic symmetry, Commun. Math. Phys. (2021, to appear) [https://arxiv.org/abs/1912.04854] [Collaborators from the University of Cambridge: R. Bauerschmidt, A. Swan] R. Bauerschmidt, T. Helmuth, A. Swan: The geometry of random walk isomorphism theorems, Ann. Henri Poincaré B (2021, to appear} [https://arxiv.org/abs/1904.01532] [Collaborators from the University of Cambridge: R. Bauerschmidt, A. Swan] R. Bauerschmidt, T. Helmuth, A. Swan: Dynkin isomorphism and Mermin-Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process, Ann. Probab. 47(5), 3375-3396, 2019. [https://arxiv.org/abs/1802.02077] [Collaborators from the University of Cambridge: R. Bauerschmidt, A. Swan] |
| Start Year | 2017 |
| Description | University of Illinois, Chicago, USA - Department of Mathematics, Statistics, and Computer Science - research collaboration |
| Organisation | University of Illinois at Chicago |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | C. Borgs, J. Chayes, T. Helmuth, W. Perkins, P. Tetali: Efficient sampling and counting algorithms for the Potts model on Z^d at all temperatures, STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, June 2020 Pages 738-751 [https://arxiv.org/abs/1909.09298] [Contributor from the University of Illinois: W. Perkins] T. Helmuth, W. Perkins, G. Regts: Algorithmic Pirogov-Sinai theory, Probab. Theory and Relat. Fields, 176, 851-895, (2020) [https://arxiv.org/abs/1806.11548] [Contributor from the University of Illinois: W. Perkins] T. Helmuth, W. Perkins, G. Regts: Algorithmic Pirogov-Sinai theory (STOC-2019), STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of ComputingJune 2019 Pages 1009-1020, [https://arxiv.org/abs/1806.11548] [Contributor from the University of Illinois: W. Perkins] |
| Start Year | 2017 |
| Description | University of Melbourne, Australia - Department of Mathematics - research collaboration |
| Organisation | University of Melbourne |
| Department | School of Mathematics and Statistics |
| Country | Australia |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | D. C. Brydges, T. Helmuth, M. Holmes, "The continuous-time lace expansion", Commun. Pure Appl. Math, (2021, to appear) [https://arxiv.org/abs/1905.09605] [Contributor from the University of Melbourne: M. Holmes] |
| Start Year | 2017 |
| Description | University of Wisconsin at Madison, USA - Department of Mathematics - research collaboration |
| Organisation | University of Wisconsin-Madison |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Collaborator Contribution | Equal partner's contribution to basic research in pure and/or mathematics. |
| Impact | S. Ledger, B. Toth, B. Valko: Random walk on the randomly oriented Manhattan lattice. Electronic Communications in Probability, 23, paper no. 43, 1-11 (2018) [arXiv:1802.01558] [Contributor from the University of Wisconsin: B. Valko] |
| Start Year | 2016 |
| Description | Alfred Renyi Centennial Conference |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | To celebrate the 100th anniversary of the birth of Alfréd Rényi the Alfréd Rényi Institute of Mathematics (Budapest) and the Hungarian Academy of Sciences jointly organize a high-profile conference held in Budapest, between 20-23 June 2022 (postponed from June 2021, due to the pandemic). Alfréd Rényi was an outstanding mathematician of the twentieth century who initiated and gave shape to many directions of modern probability, dynamical systems, ergodic theory, information theory, stochastic processes, etc. The character of the conference will reflect these aspects. The scientific program of the conference will consist of 11 plenary lectures delivered by the most distingiushed contemporary mathematicians (two Fields Medallists included and the others also from the same league) and another 27 invited lectures arranged in 9 thematic sessions. Balint Toth (PI) is member of the scientific program committee and co-chair of the organizing committee of this event. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://conferences.renyi.hu/renyi100/home |
| Description | Heilbronn Institute Workshop 2018 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Workshop on "High-dimensional critical phenomena in random environments" was held at the University of Bristol, Sep. 24--26, 2018. Participants came from the UK, Switzerland, France, Japan, Canada and the USA. This was a high profile event with some of the world leading authorities of the field of mathematical statistical physics taking part (like David Brydges and Gordon Slade). Recent progress in phase transitions in models with continuous symmetries, renormalisation group methods, random walks with long memory and many more have been discussed. The workshop was organised by Dr Tyler Helmuth and sponsored by the Heibronn Institute for Mathematical Research. |
| Year(s) Of Engagement Activity | 2018 |
| URL | http://www.tylerhelmuth.net/workshop-2018 |
| Description | Laplacians, Walks, Condensation 2017 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This was an internationally attended workshop on recent advancements in spectral analysis of Laplacians on various geometric settings and their relations to problems of diffusion and random walks, and to quantum statistical physics of spin and bosonic particle systems. Participation was conditional on invitation. World leaders in these topics contributed with more than twenty on hour long invited lecures and there was plenty of time and opportunity left for interaction and collaboration in smaller groups. The main organiser of the event was Balint Toth. |
| Year(s) Of Engagement Activity | 2017 |
| URL | http://www.bristol.ac.uk/maths/research/highlights/lrwqs-network/events/2017/lrwqs-mvdb.html?utm_sou... |
| Description | Oberwolfach Workshop 2016 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This one was the 2016 edition of the series of workshops on "Large Scale Stochastic Dynamics" held at the Oberwolfach Mathematical Research Institute (Mathematisches Forschungsinstitute Oberwolfach. MFO). The series is held once every three years and it is considered one of the most important events in this research area. Participation is by invitation only. World leaders and outstandingly gifted younger reserchers have the opportunity to present their most recent and most relevant work. This particular series of workshops has a long history (dating back to the mid-1980-s) and high prestige. Organisers: T. Bodinenau (Paris), F. Toninelli (Lyon), B. Toth (Bristol). |
| Year(s) Of Engagement Activity | 2016 |
| URL | https://www.mfo.de/occasion/1646/www_view |
| Description | Oberwolfach Workshop 2019 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This one was the 2019 edition of the series of workshops on "Large Scale Stochastic Dynamics" held at the Oberwolfach Mathematical Research Institute (Mathematisches Forschungsinstitute Oberwolfach. MFO). The series is held once every three years and it is considered one of the most important events in this research area. Participation is by invitation only. World leaders and outstandingly gifted younger reserchers have the opportunity to present their most recent and most relevant work. This particular series of workshops has a long history (dating back to the mid-1980-s) and high prestige. Organisers: T. Bodinenau (Paris), F. Toninelli (Lyon), B. Toth (Bristol). |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://www.mfo.de/occasion/1938/www_view |
| Description | Oberwolfach Workshop 2022 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This one is the 2022 edition of the series of workshops on "Large Scale Stochastic Dynamics" held at the Oberwolfach Mathematical Research Institute (Mathematisches Forschungsinstitute Oberwolfach. MFO). The series is held once every three years and it is considered one of the most important events in this research area. Participation is by invitation only. World leaders and outstandingly gifted younger reserchers have the opportunity to present their most recent and most relevant work. This particular series of workshops has a long history (dating back to the mid-1980-s) and high prestige. Organisers: P. Caputo (Rome), F. Toninelli (Vienna), B. Toth (Bristol). |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.mfo.de/occasion/2237/www_view |
| Description | Probability Seminar at Bristol |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Professional Practitioners |
| Results and Impact | This is the regular Probability Seminar held at the School of Mathematiocs, University of Bristol. In the years Between October 2017 and June 2020 Tyler Helmuth (PDRA supported by EPSRC) is the organizer of these regular events. This is a vigorous and extremely high quality seminar. The speakers form a healthy mixture of well established world leaders and early career researchers who have already proved their professional strength. |
| Year(s) Of Engagement Activity | 2018,2020 |
| URL | https://www.bristolmathsresearch.org/events/probability/ |
| Description | Probability, Analysis and Dynamics conference 2018 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The "Probability, Analysis and Dynamics -- 2018" (aka PAD2018@Bristol) was a high profile international conference, held in Bristol between 4-6 April 2018. The program of the conference consisted of 18 invited 50-minute lectures which represented some of the main aspects and developments in the intersection of the three main topics mentioned in the title. We invited speakers who are not only leading figures in these themes but are also capable of conveying their message to the audience coming from various regions of these fields. Beside the leading senior figures we also invited as speakers a smaller number of younger colleagues who have already shown their lion's claws. The chair of the oirganising committee of the conference was Balint Toth and the event was fully sponsored by the Heilbronn Institute for Mathematical Research. A similar event, "Probability, Analysis and Dynamics -- 2022" (aka PAD2022@Bristol) will be held between 6-8 April 2022. This one was originally scheduled for April 2021 but, dud to the pandemic restrictions we were forced to postpone it by one year. The chair of the oirganising committee is Balint Toth and the event is being sponsored by the Heilbronn Institute for Mathematical Research and by the Clay Mathematics Institute. This one will be already the third in the series, the previous ones were held in 2014, and in 2018. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://people.maths.bris.ac.uk/~mb13434/pad18/ |
| Description | Probability, Analysis and Dynamics conference 2022 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The "Probability, Analysis and Dynamics -- 2022" (aka PAD2022@Bristol) is a high profile international conference, wich will be held in Bristol between 6-8 April 2022. The program of the conference consists of 16 invited 50-minute lectures which represent some of the main aspects and developments in the intersection of the three main topics mentioned in the title. The conference will be of hybrid format with cca half of the lectures given in person and another half on-line. We invited speakers who are not only leading figures in these themes but are also capable of conveying their message to the audience coming from various regions of these fields. Beside the leading senior figures we also invited as speakers a smaller number of younger colleagues who have already shown their lion's claws. The co-chair of the organising committee of the conference is Balint Toth and the event is sponsored by the Heilbronn Institute for Mathematical Research, the Clay Mathematical Institute,and the London Mathematical Society. This conference was originally scheduled for April 2021 but, due to the pandemic restrictions we were forced to postpone it by one year. This one is already the third in the series, the previous ones were held in 2014, and in 2018. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://people.maths.bris.ac.uk/~mb13434/pad21/ |
| Description | Uniqueness and Algorithms in Statistical Physics |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The workshop 'Uniqueness Methods in Statistical Mechanics: recent development and algorithmic applications' took place online in Dec. 2020, and was an absolute success. The central subject of the event was on the intersection of statistical physics, computer science, probability and combinatorics. There were over 100 participants from more than 50 institutions worldwide. It was such a success that the organisers plan to make it a regular annual event. Organisers: Tyler Helmuth (Bristol/Durham) |
| Year(s) Of Engagement Activity | 2020 |
| URL | http://willperkins.org/uniqueness2020.html |