• Monte Carlo methods
• critical phenomena
• Causal Dynamical Triangulations
• Asymptotic Safety
• phase transition
• Quantum Gravity

Causal Dynamical Triangulations (CDT) is a non-perturbative approach to Quantum Gravity, based on a lattice regularization of the Einstein-Hilbert (EH) theory. The main purpose of the CDT program is to validate the Asymptotic Safety (AS) conjecture, proposed by S. Weinberg. Standard perturbative methods fail to provide a renormalizable theory of gravity. Still, in the AS scenario, it is supposed the existence of a non-trivial fixed point (in the sense of renormalization group transformations) in the parameter space around which it is possible to renormalize the EH action. CDT action is a regularized version of the EH action and it is used to sample configurations of the associated statistical system by means of Markov chain Monte Carlo methods. In the CDT phase diagram one can distinguish several different phases, namely A, B, CdS and Cb. The aim is to identify critical points where the correlation length diverges. Then, the continuum theory may possibly be found investigating the continuum limit toward these points. CDT configurations are given by triangulations where D-dimensional simplices glued together represent elementary blocks of flat space-time. A peculiar feature that distinguishes CDT from other similar approaches is the definition of a space-time foliation, where (D-1)-dimensional sub-simplices taken "at the same time" form the so-called spatial slices. Spatial slices are made of equal elements and therefore, considering the adjacency relations between them, it is possible to build a graph representation of spatial slices.

In a previous work, a new set of observables has been defined, based on the Laplace Beltrami spectral analysis. This method is a generalized version of the Fourier analysis and it is applied to graphs dual to the spatial slices of CDT configurations. Oscillation modes on the graphs give information about the properties of the system for different typical length scales. This is an important improvement with respect to the standard CDT analysis, where mainly global properties were considered in the description of the different phases. A really interesting result is the observation in some slices of a gap in the spectra of the normal modes. The spectral gap can be used to distinguish the different phases of the CDT phase diagram. With a view to the search for critical points, the Cb|CdS phase transition is the most promising one. Data suggest that there is no spectral gap (in the thermodynamical limit) in the CdS phase, while it is observed that in the Cb phase two different kinds of slices are realized: dS-type slices characterized by the absence of the spectral gap and B-type slices characterized, on the contrary, by a non-zero value of the spectral gap. Observations about the behaviour of the spectral gap near the transition line prove useful to better characterize the transition.

Results of this kind of analysis are the subject of the discussion. The spectral gap has the dimension of a mass-squared and then it defines a typical length-scale of the system. If the spectral gap vanishes in a continuous way, the related length-scale diverges approaching the transition line. Actually, this scenario seems to be realized and an estimation of the critical exponent that regulates the behaviour of the observables involved has been made. Another interesting result is the observation of a more complex discrete structure on the lower part of the spectra of B-type slices in Cb phase. This structure defines a whole hierarchy of different length-scales. Their behaviour is expected to be consistent with the considerations made above and there are encouraging results also in this direction.

Further analysis is necessary to confirm our results. Investigations on regions closer and closer to the transition line are needed but can be performed only with great numerical efforts.
Also, a more detailed study of the properties of CDT is expected to lead us to a more and more precise characterisation of slices in terms of geometrical features. Finally, CDT starting points are the EH action and the configuration space formed by foliated triangulations. Nevertheless, Laplace-Beltrami spectral methods may provide useful tools also in more general cases (general triangulations without foliated structure, higher derivative terms in the action, the presence of matter fields, etc.).