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Critical curve of two-matrix models $ABBA$, $A\{B,A\}B$ and $ABAB$, Part I: Monte Carlo

Carlos I. Pérez Sánchez

Abstract

For a family of two-matrix models \[ \frac{1}{2} \operatorname{Tr}(A^2+B^2) - \frac{g}{4} \operatorname{Tr}(A^4+B^4) - \begin{cases} \frac{h}{2} \operatorname{Tr}( A BA B) \\ \frac{h}{4} \operatorname{Tr}( A BA B+ ABBA ) \\ \frac{h}{2} \operatorname{Tr}( A B BA ) \end{cases} \] with hermitian $A$ and $B$, we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the $(h,g)$-plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.

Critical curve of two-matrix models $ABBA$, $A\{B,A\}B$ and $ABAB$, Part I: Monte Carlo

Abstract

For a family of two-matrix models with hermitian and , we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the -plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.

Paper Structure

This paper contains 17 sections, 33 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Two-matrix models
  3. Positivity
  4. Schwinger-Dyson Equations
  5. Hamiltonian Markov Chain Monte Carlo
  6. Generation of new proposals
  7. Update rules
  8. New, dynamic methods in the Python code
  9. Midpoint search
  10. Radial search
  11. Angular search
  12. Schwinger-Dyson equations and thermalisation
  13. Results
  14. Bonus: On multimatrix functional renormalisation
  15. Conclusions
  16. ...and 2 more sections

Figures (18)

  • Figure 1: Sketch of the phase diagram of three models. The top left plot lies inside the three darkened regions of each of the three other plots.
  • Figure 2: Using a fictive potential $S(X)$ (blue curve), rules $R_1$ and $R_2$ are sketched. In each situation, the beads on each arrow tail mean the last Markov chain member, and at the arrow tip the proposed move $\tilde{X}$. If $\Delta S<0$, according to $R_1$, one accepts the move. For $R_2(p)$ in the left-top case, the increment in $S$ leads to rejection, since $N \Delta S> \log 1/p$, whereas the $R_2(\tilde{p})$ is verified in the case of the proposed move for the purple points, as the bound $(1/N) \log (1/\tilde{p})$ was not exceeded ($p,\tilde{p}$ are freshly uniformly chosen random numbers in $(0,1)$ for each acceptance test).
  • Figure 3: On the Hamiltonian approach to MCMC with the leapfrog integrator. The left (right) plane is the 'configuration' (resp. momenta) $\mathrm{H}_N^2$-space of matrices $X_t=(A_t,B_t)$ [resp. $P_{t+1/2}=(P_{A,t+1/2},P_{B,t+1/2})$].
  • Figure 4: A naive test of lattice points is not efficient. This leads to dynamical methods to search the boundary of the convergent region, see in Sec. \ref{['sec:Methods']}.
  • Figure 5: 'Expectation (a) vs. reality (b)'. In (a), instead of $\texttt{MC}$ we use as dummy a characteristic function. When points of opposite truth value lie a fixed distance $\delta$ apart (a 'dipole') their middle point is added to the critical curve.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Example 2.1