Research Interests

I’m interested in the mathematics of statistical mechanics and statistical field theory, and the connections of these topics with other branches of mathematics (e.g., mathematical physics; probability theory; theoretical computer science; combinatorics). Some specific topics I have worked on include: spin systems; efficient sampling algorithms; self-interacting random walks; the arboreal gas; the lace expansion; fermionic and supersymmetric methods.

Publications

  • S. Cannon, T. Helmuth, W. Perkins, “Pirogov—Sinai Theory Beyond Lattices”, preprint, 40 pp. [arXiv]

  • A. Hammond, T. Helmuth, “Directed Spatial Permutations on Asymmetric Tori”, Ann. Probab., to appear. 31 pp. 2024. [arxiv]

  • T. Helmuth, R. Mann, “Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature“, Quantum, 12 pp. 2023. [arXiv]

  • R. Bauerschmidt, T. Helmuth, “Spin systems with hyperbolic symmetry: a survey”, Proceedings of the ICM 2022, 17 pp. [arXiv]

  • T. Helmuth, H. Lee, W. Perkins, M. Ravichandran, and Q. Wu, “Approximation algorithms for the random-field Ising model”, SIAM Journal on Discrete Mathematics, 37(3), 1610-1629. 2023. [arXiv]

  • R. Bauerschmidt, N. Crawford, T. Helmuth, “Percolation transition for random forests in d≥3”, Invent. Math., 1-96. 2024. [arXiv]

  • T. Helmuth, M. Jenssen, W. Perkins, “Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs”, Ann. Henri Poincaré B, 59(2), 817-848. 2023. [arXiv]

  • T. Helmuth, R. Mann, “Efficient algorithms for approximating quantum partition functions”, J. Math. Phys., 62(1), 022201. 2021. [arXiv]

  • T. Helmuth, A. Shapira, “Loop-erased random walk as a spin system observable”, J. Stat. Phys., 181(4), 1306-1322. 2020. [arXiv]

  • T. Helmuth, W. Perkins, S. Petti, “Correlation decay for hard spheres via Markov chains”, Ann. Appl. Probab., 32(3), 2063-2082. 2022. [arXiv]

  • R. Bauerschmidt, N. Crawford, T. Helmuth, A. Swan, “Random spanning forests and hyperbolic symmetry”, Commun. Math. Phys., 381, 1223-1261. 2021. [arXiv]

  • 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”, Random. Struct. Algorithms, 63, 130-170. 2023. [arXiv]

    An extended abstract of this work appeared in STOC 2020. 

  • D. C. Brydges, T. Helmuth, M. Holmes, "The continuous-time lace expansion", Commun. Pure Appl. Math., 74, 2251-2309. 2021. [arXiv]

  • R. Bauerschmidt, T. Helmuth, A. Swan, “The geometry of random walk isomorphism theorems”,

    Ann. Henri Poincaré B, 57(1), 408-454, 2021. [arXiv]

  • T. Helmuth, W. Perkins, G. Regts, "Algorithmic Pirogov—Sinai theory", Probab. Theory and Relat. Fields, 176, 851-895, 2020. [arXiv]

    An extended abstract of this work appeared in STOC 2019.

  • 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. [arXiv]

  • A. Hammond, T. Helmuth, "Self-attracting self-avoiding walk", Probab. Theory and Relat. Fields, 175(3-4), 677-719. 2019. [arXiv]

  • T. Helmuth, "Dimensional reduction for generalized continuum polymers", J. Stat. Phys., 165(24), 24-43. 2016. [arXiv]

  • T. Helmuth, "Loop-weighted walk", Ann. Henri Poincaré D., 3, 55-119. 2016. [arXiv]

  • T. Helmuth, "Ising model observables and non-backtracking walks", J. Math. Phys. 55(8), 1--28. 2014. [arXiv]

  • T. Helmuth, G. Patrick, "Two rolling disks or spheres", Discrete Contin. Dynam. Systems Ser. S 3(1), 129--140. 2009.

  • T. Helmuth, R. Spiteri, J. Szmigielski, "One-dimensional magnetotelluric inversion with radiation boundary conditions", Canad. Appl. Math. Quart. 15(4), 419--444. 2007.