The spectral dimension of 2D quantum gravity

Abstract

We show that the spectral dimension d s of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c ≤ 1.The same arguments provide a simple proof of the known result d s = 4/3 for branched polymers.

Liouville heat kernel: Regularity and bounds

“…We emphasize that the exponent 1/(1 + γ 2 /4) is not expected to be optimal, as in our derivation we do not take into account the geometry of the Gaussian field. 1 For γ 2 8/3, the same heat kernel lower bound holds when the endpoints are sampled according to the measure M γ . This is proven in Section 5.3 for γ 2 ≤ 4/3 and extended to γ 2 ≤ 8/3 in Section 5.4.…”

“…, called the spectral dimension of LQG, is equal to 2: this has been heuristically computed by Ambjørn et al in [1] and then rigorously derived in a weaker form in [37]. This yields the relation…”

“…It has been suggested by Watabiki [41,1] that the (conjectural) metric space (T, d γ ) is locally monofractal with intrinsic Hausdorff dimension 2) (note that in the special case of pure gravity γ = 8 3 this gives d H (γ) = 4 which is compatible with the dimension of the Brownian map). This could suggest that the heat kernel p γ t (x, y) takes the following form for small t:…”

Random walks on combs

“…The spectral dimension of random surfaces is believed to be 2 [5,8] while the Hausdorff dimension is known to be 4 [24], see also [25,26]. It has been shown recently [25,26] that the generic structure of infinite planar random surfaces (triangulations) is analogous to that of the random infinite trees discussed in Section 5.…”

Recurrence of Distributional Limits of Finite Planar Graphs

“…From a mathematical viewpoint, natural measures that were considered are the uniform measures on isomorphism classes of triangulations of the sphere with a fixed number of vertices. In [AAJ+98,ANR+98] diffusion on some random surfaces and random walks on random triangulations including the uniform measure have been considered. It was suggested there that the probability for the random walk to be at time t at its starting vertex should decay like t-I , provided that t is not too large relative to the size of the triangulation, and that the mean square displacement at time t is t l / 2 .…”