Bootstraps to strings: solving random matrix models with positivity

Abstract

A new approach to solving random matrix models directly in the large N limit is developed. First, a set of numerical values for some low-pt correlation functions is guessed. The large N loop equations are then used to generate values of higher-pt correlation functions based on this guess. Then one tests whether these higher-pt functions are consistent with positivity requirements, e.g., tr M 2k ≥ 0. If not, the guessed values are systematically ruled out. In this way, one can constrain the correlation functions of random matrices to a tiny subregion which contains (and perhaps converges to) the true solution. This approach is tested on single and multi-matrix models and handily reproduces known solutions. It also produces strong results for multi-matrix models which are not believed to be solvable. A tantalizing possibility is that this method could be used to search for new critical points, or string worldsheet theories.

Bootstrapping Matrix Quantum Mechanics

“…Our work is directly inspired by a recent beautiful paper by Lin [18], with a similar approach also being employed in Ref. [19].…”

Bootstrap bounds on closed Einstein manifolds

“…This gives general constraints on the possible values that the observables can take. Examples of interesting physical systems that have been recently studied using such methods include strongly coupled conformal field theories (CFTs) [1], scattering amplitudes [2,3], and random matrix models [4]. These studies JHEP10(2020)069…”

“…Triple overlap integrals such as these are natural objects associated to any closed manifold. 4 We can also expand products involving derivatives of eigenfunctions,…”

“…We obtain an upper bound on λ a 3 /λ a 1 for each λ a 2 /λ a 1 ∈ (0, 4] as shown in figure 16, where we take λ i ≥ 0 and either λ I ≥ 0 (solid lines) or λ I /λ a 1 ≥ 4 (dashed lines). The very long flat N -tori at position (4,9) saturate the λ I ≥ 0 bounds, as marked by the cross, and the round spheres lie at the edge of the λ I /λ a 1 ≥ 4 bounds, as shown by the filled circles.…”

“…The converse is not true since infinitesimal Einstein deformations do not always integrate to curves of Einstein structures[23] 4. We expect that generically the triple overlap integrals should be non-zero unless there is a symmetry that causes them to vanish.…”