• spectral methods
• monte carlo methods
• asymptotic safety
• quantum gravity
• causal dynamical triangulations

Different aspects of the problem of formulating a consistent and predictive theory
of Quantum Gravity are still open.
Nevertheless, in the last few decades, many approaches have been proposed
and the research on quantum gravity is now proceeding by exploring
connections and obtaining new insights from the cross-fertilization between them.
A family of approaches is drawing more and more interest recently:
these are based on usual formulations of gravity as the Einstein-Hilbert theory or generalizations,
but the mechanism which would make renormalization possible
relies on the existence of a nonperturbative point in the space of couplings where
the theory is well defined even at arbitrarily small scales (ultraviolet),
regime which seems inaccessible to standard perturbative techniques.
The occurrence of such non-trivial point in the space of parameters
is called the \emph{asymptotic safety scenario},
which can be investigated by means of different techniques,
such as functional renormalization group approaches and numerical approaches
based on Monte Carlo integration of the path-integral over geometries.
In the latter class belongs the approach known as Causal Dynamical Triangulations (CDT),
for which the geometries are approximated by triangulations (also called simplicial manifolds), i.e., manifolds composed of elementary flat building blocks of finite size (simplexes) and with a causal condition of global hyperbolicity that enforces a globally well-defined cosmological time.

In the CDT approach, where the path-integral is formulated
in terms of a Monte Carlo sampling of configurations from an appropriate statistical distribution,
asymptotic safety can be tested by searching for a point in the phase diagram
of the equivalent statistical system where a second-order transition
with diverging correlation length occurs, and for which the continuum limit can be investigated.
In literature, there is strong support for the existence of two second-order lines
in the phase diagram, but whether one of these points corresponds to a continuum theory of quantum gravity consistent and has the expected semiclassical behavior has still to be established.

The first part of this thesis deals with the problem of defining useful observables for the study of critical properties in CDT, for which we present and investigate new techniques based on the analysis of eigenvalues and eigenvectors of the Laplace--Beltrami (LB) operator on
simplicial manifolds.
Indeed, the solutions to the LB operator eigenproblem define what is the Fourier transform
on the manifold under investigation,
and allow us to classify hierarchically the characteristic length scales of the geometries,
in particular their large scale (relevant) properties.

We start by considering the spectrum of dual graph representation of triangulations.
This representation encodes information about the adjacency relations between simplexes, and, from these relations, one can build the Laplace matrix,
which acts as an approximation to the LB operator.
From the spectrum of the Laplace matrix we can extract some useful quantities;
in particular, the smallest non-zero eigenvalue $\lambda_1$, also called \textit{spectral gap}
gives us information about the largest characteristic length
and connectivity properties of the geometries, while the \textit{effective dimension} $d_{EFF}$ tells us how the characteristic lengths change when observed at different scales.
By applying the spectral graph analysis to the spatial slices of CDT triangulations,
we first identify the spectral characteristics of the four phases of the phase diagram
and then we investigate the critical behavior of the transition line between the so-called
$C_{b}$ and $C_{dS}$ phases (a promising candidate for a consistent continuum limit) by employing the spectral gap as order parameter for the transition, vanishing only in the $C_{dS}$ phase.

However, in the effort of generalizing the spectral graph analysis to full-fledged four-dimensional causal dynamical triangulations, we run into a couple of issues with the dual graph representation.
We analyze the origin of these issues, and propose, as solution, another representation
for the LB operator, coming from the framework of Finite Element Methods (FEM)
and based on a weak formulation of the eigenproblem.
Using FEM, which come backed up by theorems guaranteeing convergence,
it is possible to set up a strategy, called refinement, that allows us to approximate with arbitrary accuracy the exact spectrum of the infinite-dimensional LB operator, while this strategy is not available for dual graphs (except in dimensions two).
We compare the two methods first on test geometries and then on spatial slices
of CDT configurations that had been already investigated using dual graphs,
and we find significant discrepancies in the estimates of both the effective dimension in the $C_{dS}$ phase and the critical index of the $C_b$-$C_{dS}$ transition.

The second part of this thesis, in the spirit of pushing toward a realistic connection
with phenomenology, involves the minimal coupling of compact gauge fields of the Yang--Mills
type with CDT, where link variables are placed in the edges of the dual graph.
Our setup us is general, but we focus in particular to an exploratory numerical investigation of the system of two-dimensional CDT minimally coupled to gauge fields with either $U(1)$ or $SU(2)$ gauge groups,providing an explicit construction for the Markov chain moves.
By studying the effects of gauge fields on gravity using as observables the total volume
and the correlation length of the distribution of spatial volumes along spatial slices
(volume profiles) we find that the backreaction of fields on the geometry trivially amounts
to a shift in the bare parameter associated to the cosmological constant.
The effects of gravity on the gauge fields, investigated by using the correlation length of flux-related quantities called torelons, turn out to be less trivial. We also study the $\theta$-dependence of the $U(1)$ case by analyzing the topological charge, which behaves like a winding number for the gauge fields, and the topological susceptibility.
Incidentally, we discover that the problem of critical slowing down, affecting different lattice field theories in the continuum limit, seems mitigated by orders of magnitude when one consider locally variable geometries, suggesting possible applications also in different contexts.