Abstract
Bootstrap is an idea that imposing consistency conditions on a physical system may lead to rigorous and nontrivial statements about its physical observables. In this work, we discuss the bootstrap problem for the invariant measure of the stochastic Ising model defined as a Markov chain where probability bounds and invariance equations are imposed. It is described by a linear programming (LP) hierarchy whose asymptotic convergence is shown by explicitly constructing the invariant measure from the convergent sequence of moments. We also discuss the relation between the LP hierarchy for the invariant measure and a recently introduced semidefinite programming (SDP) hierarchy for the Gibbs measure of the statistical Ising model based on reflection positivity and spin-flip equations.
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References
J. Glimm and A. Jaffe, Quantum Physics — A Functional Integral Point of View, Springer, New York, U.S.A. (1987).
T. Liggett, Interacting Particle Systems, Classics in Mathematics, Springer, Heidelberg, Germany (2004).
L. Onsager, Crystal statistics. 1. A two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944) 117 [INSPIRE].
T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Spin spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B 13 (1976) 316 [INSPIRE].
K. Binder and D.W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer, Heidelberg, Germany (2010).
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) Models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
M. Cho, B. Gabai, Y.-H. Lin, V.A. Rodriguez, J. Sandor and X. Yin, Bootstrapping the Ising Model on the Lattice, arXiv:2206.12538 [INSPIRE].
P.L. Dobruschin, The description of a random field by means of conditional probabilities and conditions of its regularity, Theory Probab. Appl. 13 (1968) 197.
O.E. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Commun. Math. Phys. 13 (1969) 194.
O. Hernández-Lerma and J. Lasserre, Markov Chains and Invariant Probabilities, Progress in Mathematics, Birkhäuser, Basel, Switzerland (2012).
M. Korda, D. Henrion and I. Mezic, Convex computation of extremal invariant measures of nonlinear dynamical systems and Markov processes, arXiv:1807.08956.
G. Fantuzzi, D. Goluskin, D. Huang and S.I. Chernyshenko, Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization, arXiv:1512.05599.
I. Tobasco, D. Goluskin and C.R. Doering, Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems, Phys. Lett. A 382 (2018) 382 [arXiv:1705.07096].
J.B. Lasserre, An explicit exact sdp relaxation for nonlinear 0-1 programs, in Integer Programming and Combinatorial Optimization, K. Aardal and B. Gerards eds., Springer, Heidelberg, Germany (2001), pg. 293.
J.B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11 (2001) 796.
M. Laurent, Sums of Squares, Moment Matrices and Optimization Over Polynomials, Springer, New York, U.S.A. (2009), pg. 157.
S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge University Press, Cambridge, U.K. (2017).
R. Holley, Free energy in a Markovian model of a lattice spin system, Commun. Math. Phys. 23 (1971) 87.
K. Schmüdgen, The k-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991) 203.
T.M. Liggett, J.E. Steif and B. Tóth, Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem, math/0512191.
V. Kazakov and Z. Zheng, Analytic and numerical bootstrap for one-matrix model and “unsolvable” two-matrix model, JHEP 06 (2022) 030 [arXiv:2108.04830] [INSPIRE].
M. ApS, The MOSEK optimization toolbox for MATLAB manual. Version 10.0, (2023).
V. Kazakov and Z. Zheng, Bootstrap for lattice Yang-Mills theory, Phys. Rev. D 107 (2023) L051501 [arXiv:2203.11360] [INSPIRE].
Acknowledgments
We greatly appreciate Clay Cordova for persistently asking M.C. about the convergence of the Ising bootstrap, which initiated this work. We would like to thank Hamza Fawzi, Weihao Guo and Zechuan Zheng for helpful discussions, and Hamza Fawzi for helpful comments on the preliminary draft. We also thank the anonymous referee of the Journal of High Energy Physics for suggestions on the draft. M.C. is supported by the Sam B. Treiman Fellowship at the Princeton Center for Theoretical Science. X.S. was partially supported by the NSF Career award 2046514, and by a fellowship from the Institute for Advanced Study at Princeton during 2022-2023.
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ArXiv ePrint: 2309.01016
On leave from the University of Pennsylvania. (Xin Sun)
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Cho, M., Sun, X. Bootstrap, Markov Chain Monte Carlo, and LP/SDP hierarchy for the lattice Ising model. J. High Energ. Phys. 2023, 47 (2023). https://doi.org/10.1007/JHEP11(2023)047
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DOI: https://doi.org/10.1007/JHEP11(2023)047