In this thesis some equilibrium and non-equilibrium statisticalmethods are implied on two different versions of non-perturbativequantum gravity.Firstly, we report a novel statistical mechanics in which a class of evolutionary mapsact on trivalent spin network in randomly chosen initial statesand give rise to Self-organized Criticality. The result of continuously applying these maps indicate an expansion in the space-time area associated.Secondly, a previously unknown statistical mechanics in quantum gravity is introduced in the framework of two dimensional Causal Dynamical Triangulations. This provides us a useful and new tools to understand this quantum gravity in terms of effective spins. This study reveals a correspondence between the statistics of Anti-ferromagneticsystems and Causal Dynamical quantum gravity. More importantly, it provides a basis for studying anti-ferromagnetic systems in a background independent way.Thirdly, two novel properties of area operator inLoop Quantum Gravity are reported: 1) the generic degeneracyand 2) the ladder symmetry. These were not known previously for years.The first one indicates thatcorresponding to any eigenvalue of area operator in loop quantum gravity there exists a finite number ofdegenerate eigenstates. This degeneracy is shown to be one way for the explanation of black hole entropy in a microscopic way. More importantly, we reproduce Bekenstein-Hawking entropy of black hole by comparing the minimal energy of a decaying frequency from a loop quantum black hole and the extracted energy from a perturbed black hole in the highly damping mode. This consistency reveals a treasure model for describing a black hole in loop quantum gravity that does nor suffer from the restrictions of an isolated horizon. The second property indicates there exists a ladder symmetry unexpectedlyin the complete spectrum of area eigenvalues. This symmetry suggests the eigenvalues of area could be classified into different evenlyspaced subsets, each called a `generation.;; All generations are evenlyspaced; but the gap between the levels in any every generation isunique. One application of the two new properties of area operatorhave been considered here for introducing a generalized picture ofhorizon whose area cells are not restricted to the subset consideredin quantum isolated horizon theory. Instead, the area cells acceptsvalues from the complete spectrum. Such horizon in the presence ofall elements of diffeomorphism group contains a number of degreesof freedom independently from the bulk freedom whose logarithmscales with the horizon area. Note that this is not the case in quantum isolated horizon when the complete elements of diffeomorphismapplies.Finally, we use a simple statistical method in which no pre-assumption is made for the essence of the energy quanta radiatedfrom the hole. We derive the effects of the black hole horizon fluctuations and reveal a new phenomenon called ;;quantum amplificationeffects;; affecting black hole radiation. This effect causes unexpectedly a few un-blended radiance modes manifested in spectrum asdiscrete brightest lines. The frequency of these modes scales withthe mass of black hole. This modification to Hawking;;s radiationindicates a window at which loop quantum gravity can be observationally tested at least for primordial black holes.
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