The Spectral Dimension of Generic Trees

Abstract

Abstract. We define generic ensembles of infinite trees. These are limits as N → ∞ of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension d h = 2. Our main result is that their spectral dimension is d s = 4/3, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.

Generic Ising trees

“…On the other hand, the same equation characterizes the random tree models considered in [13] except for a rescaling of the coupling constant g by the factor 2 cosh β. It follows that the condition (39) can be considered as a generalization of the genericity condition introduced in [13]. For this reason, the results on the Hausdorff dimension and the spectral dimension established in this paper follow from [13] in the case h = 0.…”

“…We will further assume the branching weights to satisfy a genericity condition explained below in (39), and which defines the generic Ising tree ensembles considered in this paper (see also [13]). Finally, the partition function Z N in (7) is given by…”

“…The models are defined in terms of branching weights p n , n ≥ 0, associated to vertices and subject to a certain genericity condition [13]. The latter is a rather mild requirement, satisfied e.g.…”

Spectral dimension flow on continuum random multigraph

“…Given that the resistance of recurrent multigraphs is infinite this inequality implies that the twodimensional UICT is recurrent and d s ≤ 2 almost surely. Furthermore it implies that the recurrent multigraph ensemble and the generic tree ensemble are two extreme cases used to bound the spectral dimension of UICT and saturate the left and right hand side of (1) respectively (note that all the above graphs have d H = 2 and d s (GT ) = 4/3 [9,13]). It is believed that the spectral dimension of UICT is two and that thus multigraphs provide a tight bound.…”

The spectral dimension of random brushes

“…If p G (t) ∼ t −ds/2 (1) as t → ∞ then we say that d s is the spectral dimension of the graph G. The existence of d s is not guaranteed for individual graphs but its ensemble average can be shown to be well defined in many cases [3,4]. It is easy to see that if the spectral dimension exists then it is independent of the starting site of the random walk.…”