Bootstraps to strings: solving random matrix models with positivity
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Springer
Received: April 30, 2020 Accepted: May 22, 2020 Published: June 15, 2020
Henry W. Lin Jadwin Hall, Princeton University, Princeton, NJ 08540, U.S.A. Google, Mountain View, CA 94043, U.S.A.
E-mail: [email protected] 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., htr M 2k i ≥ 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. Keywords: Matrix Models, 1/N Expansion, Field Theories in Lower Dimensions ArXiv ePrint: 2002.08387
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)090
JHEP06(2020)090
Bootstraps to strings: solving random matrix models with positivity
Contents 1
2 The loop equations 2.1 The search space
3 3
3 Positivity constraints 3.1 Positivity for one matrix ensembles 3.1.1 Relation to the Hamburger moment problem 3.2 Multi-matrix models and the general algorithm
5 5 6 6
4 Bootstrapping 1-matrix models 4.1 Single Hermitian matrix 4.1.1 Unbounded potentials and the tip of the peninsula 4.1.2 Other single-matrix models
7 7 11 12
5 Bootstrapping multi-matrix models
14
6 Discussion
16
A Review of the single matrix model A.1 Single cut A.2 Multi-cut solutions
18 18 19
B The bootstrap approach for computing determinants or vectors
20
C The bootstrap approach for 1/N corrections
21
D Ising model on a random planar lattice D.1 Relation to the cubic interaction
22 24
E Mathematica code for generating loop equations
26
1
Introduction
From ancient days, sages appreciated that certain large N theories simplify dramatically. For matrix theories in the ’t Hooft limit, one needs to sum only a tiny subset of all possible Feynman diagrams, the planar ones [1, 2]. However, with a few notable exceptions, this simplification is not enough: even in the ’t Hooft limit, most matrix theories are impossible to solve. This is true even for zero dimensional statistical ensembles of a small number of
–1–
JHEP06(2020)090
1 Introduction
matrices. With the exception of the single matrix model [3–6], almost all models remain unsolved.1 In this paper, we propose a method to solve multi-matrix models in the strict large N limit. For our purposes, a 2-matrix model is defined by an integral of the form Z = lim
N →∞
Z
dA dB e−N Tr
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