Goldilocks and the bootstrap

• Published in: The journal of high energy physics Vol. 2025; no. 9; pp. 109 - 35
• Main Authors: Berenstein, David, Rodriguez, Victor A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 12.09.2025; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Algorithms and Theoretical Developments; Applied mathematics; Classical and Quantum Gravitation; Eigenvalues; Elementary Particles; Exact solutions; Integrals; Lattice Quantum Field Theory; Matrix Models; Particle physics; Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Statistical analysis; Statistical methods; String Theory; Symmetry; t Hooft and Polyakov loops; Wilson; Lattice Quantum Field Theory; Wilson; Algorithms and Theoretical Developments; t Hooft and Polyakov loops; Matrix Models
• ISSN: 1029-8479; 1126-6708; 1127-2236; 1029-8479
• DOI: 10.1007/JHEP09(2025)109

A bstract We study simplified bootstrap problems for probability distributions on the infinite line and the circle. We show that the rapid convergence of the bootstrap method for problems on the infinite line is related to the fact that the smallest eigenvalue of the positive matrices in the exact solution becomes exponentially small for large matrices, while the moments grow factorially. As a result, the positivity condition is very finely tuned. For problems on the circle we show instead that the entries of the positive matrix of Fourier modes of the distribution depend linearly on the initial data of the recursion, with factorially growing coefficients. By positivity, these matrix elements are bounded in absolute value by one, so the initial data must also be fine-tuned. Additionally, we find that we can largely bypass the semi-definite program (SDP) nature of the problem on a circle by recognizing that these Fourier modes must be asymptotically exponentially small. With a simple ansatz, which we call the shoestring bootstrap, we can efficiently identify an interior point of the set of allowed matrices with much higher precision than conventional SDP bounds permit. We apply this method to solving unitary matrix model integrals by numerically constructing the orthogonal polynomials associated with the circle distribution.