Universal Properties of Spectral Dimension

Truncation effects in the functional renormalization group study of spontaneous symmetry breaking

“…this way one can explicitly show that for N ≥ 2 there is no SSB for d ≤ 2, with d being real, while SSB occurs for d > 2 [9]. In [10] the study of how O(N ) universality classes depends continuously on the dimension d (and as well on N ), in particular for 2 < d < 3, was recently presented.…”

Crafting networks to achieve, or not achieve, chaotic states

“…We shall show that the rewiring of a few links, beyond altering significantly the network structure, can also entail different collective behaviors: in particular, we shall investigate if, like on regular lattices, we have a spontaneous symmetry breaking for d > 2, which is absent when d < 2. This brings some analogies to the extension of the Mermin-Wagner theorem on inhomogeneous structures [22,23] in which the critical parameter to discriminate between different regimes is the spectral dimension [24][25][26], therefore opening an interesting thread of research. Moreover we shall focus on d = 2, or r ∼ √ N , to see if a chaotic state emerges, displaying some similarities woth the one observed in the regular structure discussed in [13].…”

The spectral dimension of random trees

“…To be precise the denomination 'spectral dimension' refers to the behaviour of the spectral density ρ(l) of low-lying eigenvalues of the Laplacian L, namely ρ(l) ∼ ld /2−1 . In fact, it can be shown [3] thatd is connected to the long-time behaviour of the graph average of P xx (t), or equivalently to the infrared power-like singularity of the graph average (13) where |B(o, r)| is the volume of B(o, r).d is therefore sometimes called the 'average' spectral dimension. On regular lattices it coincides with the local spectral dimension and the usual Euclidean dimension.…”

“…On macroscopically inhomogeneous structures this average dimension may differ from the local one, which enters the long-time tail of the random walk return probability on a given node [3]. However, it is the average spectral dimension that plays the same role of the lattice Euclidean dimension in many contexts, as for instance in providing a consistent criterion on whether a continuous symmetry breaks down at low temperatures [4].…”

“…To be precise, we refer here to the average spectral dimension, that is a global property of the graph, related for instance to the infrared singularity of the Gaussian process or, equivalently, to the density of small eigenvalues of the Laplacian or the graph average of the long-time tail of the random walk return probability [3]. On macroscopically inhomogeneous structures this average dimension may differ from the local one, which enters the long-time tail of the random walk return probability on a given node [3].…”