using a lattice approximation and then calculate the observables over that set of points [@lattice]. Since an approximation in the position space is not suitable due to the oscillator factor of the Moyal product, instead the lattice approximation, will be used the matrix Moyal base, which was already used in the first renormalization proof of $\varphi^4$-model restricted to finite matrices. Hence, will be performed a Monte Carlo simulation studying some statistical quantity such the energy density and specific heat varying the parameters $\Omega, \frac{\chi_{-1}}{\chi_0}, \alpha$ and gathering some informations on the various contributions of the fields to the action. The simulations are quite cumbersome due the complexity of the action and the number of independent matrices to handle but we are able to get an acceptable balance between the computation precision and the computation time. For the simulations is applied a standard Metropolis-Monte Carlo algorithm [@Metro] with various estimators for the error and for the autocorrelation time of the samples. In general we chose the range of parameters in order to avoid problems with the thermalization process, obtaining numerical simulations where is enough to wait a relative small number of Monte Carlo steps to compute independent results from the initial 