Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

Abstract

We study a matrix model that has φ i a (a = 1, 2, . . . , N, i = 1, 2, . . . , 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 R ∼ 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.

1/N expansion of circular Wilson loop in $$ \mathcal{N} $$ = 2 superconformal SU(N) × SU(N) quiver

“…which is thus consistent with the analytic estimate (6.1). 15 For other papers using MC methods in matrix models see, e.g., [36][37][38][39][40][41] and section 6.5 of [42]. In the explored region of parameter space the Metropolis-Hastings algorithm turned out to be faster than the Hybrid Monte Carlo one, taking into account autocorrelation.…”

Numerical and analytical studies of a matrix model with non-pairwise contracted indices

“…A matrix model with non-pairwise contracted indices Naoki Sasakura necklace diagrams depicted in Figure 2, where the examples of the order n = 3 are given. Summing over all n, it has been obtained that A generalization of the perturbative computation above was performed in [19] by employing a slightly different strategy. In the generalization, the coupling term in the exponent was generalized to…”

“…The Monte Carlo simulations were performed by the Metropolis update method in [19]. We observed a phase transition or a crossover around R ∼ N 2 /2.…”

Counting Tensor Rank Decompositions

“…There, we derive a closed formula for Z R (∆), which is guaranteed to exist under the condition that a certain quantity G R , which is independent of ∆, exists and is finite. Another interesting connection to the CTM is found at this point, since this quantity G R is a generalisation of the partition function of the matrix model studied in [14][15][16]. In Section 4, the existence of G R is proven for R = 1, and numerical analysis is conducted for R > 1 for a specific choice of volume form dΦ to arrive at a conjecture for the maximal allowed value of R, called R c .…”

“…A noteworthy fact about the CTM is that it has several known exact solutions for the quantum constraint equations [11]. One of these has recently been extensively analysed due to the emergence of Lie group symmetries in this wave function, which potentially hints towards the emergence of macroscopic spacetimes [12][13][14][15][16][17]. This wave function, in the Q-representation, is closely related to a statistical model [17] that is mathematically equivalent to…”