Lorenzo Grimaldi — Università di Roma "Tor Vergata" e Centro Ricerche "Enrico Fermi" #
Entropy and heat capacity of regular fractal graphs #
The Laplacian Renormalisation Group (LRG) has been introduced as a means to generalise the usual coarse-graining procedure of homogeneous systems to heterogeneous networks. It draws on the Gaussian model of random graphs and the diffusion equation and treats the diffusion time and the Laplacian operator as the analogous of the inverse of the temperature and the Hamiltonian, respectively. This defines a network propagator and an ensemble of accessible configurations, in analogy with the canonical ensemble of statistical physics. The construction yields natural generalisations of entropy and heat capacity to the case of heterogeneous networks, thus providing us with the tools to study possible phase transitions and to identify the optimal scale at which the coarse-graining can be performed. Recent developments have displayed the ability of the LRG to unveil information about the structure of the considered network. Whenever the heat capacity displays a plateau over a time interval, the system becomes scale-invariant within that interval and the value of the heat capacity yields the spectral dimension of the underlying geometry. Moreover, it has been shown that the Fiedler eigenvalue scales as a power of the number of nodes of the system, the value of the index depending on the specific geometry. In some cases- like the Sierpinski gasket and carpet- such value is again related to the spectral dimension of the graph. We examined the behaviour of some regular fractal graphs- namely variations of the gasket and carpet. Within these frameworks, we observed that the LRG machinery produces the correct value of the spectral dimension of the graph, both from the perspective of the heat capacity and that of the Fiedler eigenvalue. On the other hand, we also analysed different configurations of the Dirac comb, for which the scaling of the Fiedler eigenvalue is not associated to the spectral dimension of the graph. Indeed, the LRG investigation succeeded also in this setting.