Séminaire de Physique ThéoriqueMatrix bootstrap revisited
Vladimir Kazakov (ENS Paris-Saclay)
Thursday 16 December 2021 15:30 - Institut Denis Poisson, Tours - la salle E1020
Matrix bootstrap is a new method for the numerical study of (multi)-matrix models in the planar limit, using loop equations for moments of distribution (Ward identities and factorization of traces at infinite N). The lack of information associated with the use of only a finite number of lower moments is supplemented by the conditions of positivity of the correlation matrix. The numerical solution of loop equations and these conditions leads to inequalities for the lowest moments, which rapidly converge to exact values with an increase in the number of used moments. In our work https://arxiv.org/pdf/2108.04830.pdf, the method was tested on the example of the standard one-matrix model, as well as on the case of an "unsolvable" 2-matrix model with the interaction tr[A, B]^2 and with quartic potentials. We propose a significant improvement of original H.Lin's proposal for matrix bootstrap by introducing the relaxation procedure: we replace the non-convex, non-linear loop equations by convex inequalities. The results look quite convincing and matrix bootstrap seems to be an interesting alternative to the Monte Carlo method. For example, for <trA ^ 2>, the precision reaches 6 digits (with modest computer resources). I will discuss the prospects for applying the method in other, physically interesting systems.