Abstract
We study a matrix model that has phi _a^i (a=1,2,ldots ,N, i=1,2,ldots ,R) as its dynamical variable, whose lower indices are pairwise contracted, but upper ones are not always done so. This matrix model has a motivation from a tensor model for quantum gravity, and is also related to the physics of glasses, because it has the same form as what appears in the replica trick of the spherical p-spin model for spin glasses, though the parameter range of our interest is different. To study the dynamics, which in general depends on N and R, we perform Monte Carlo simulations and compare with some analytical computations in the leading and the next-leading orders. A transition region has been found around Rsim N^2/2, which matches a relation required by the consistency of the tensor model. The simulation and the analytical computations agree well outside the transition region, but not in this region, implying that some relevant configurations are not properly included by the analytical computations. With a motivation coming from the tensor model, we also study the persistent homology of the configurations generated in the simulations, and have observed its gradual change from S^1 to higher dimensional cycles with the increase of R around the transition region.
Highlights
The criterion above can in principle be checked by studying the properties of a wave function of each theory
We studied a matrix model containing nonpairwise index contractions [21], which has a motivation from a tensor model of quantum gravity [16,17]
It has φai (a = 1, 2, . . . , N, i = 1, 2, . . . , R) as its degrees of freedom, where the lower indices are pairwise contracted, but the latter are not always done so. This matrix model has the same form as what appears in the replica trick of the spherical p-spin model for spin glasses [27,28], though the variable and parameter ranges of our interest are different
Summary
The criterion above can in principle be checked by studying the properties of a wave function of each theory. To simplify the problem keeping the main structure from the tensor model as much as possible, one of the authors of the present paper and his collaborators considered the following two simplifications in the former papers One is that they considered a toy wave function rather than the actual. While this substantially simplifies the analysis, the toy wave function keeps the most crucial property mentioned above that there appear peaks at the tensor values that are symmetric under Lie groups as the actual wave function of the tensor model does [20] Though this toy wave function is simpler than the actual one, it is still difficult to perform thorough analyses, because the dimension of the argument (a symmetric tensor with three indices) of the wave function is very large with the order of ∼ N 3/6. In “Appendix F”, a brief introduction to persistent homology is given
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