The spectral dimension of the branched polymer phase of two-dimensional quantum gravity

Abstract

The metric of two-dimensional quantum gravity interacting with conformal matter is believed to collapse to a branched polymer metric when the central charge c > 1. We show analytically that the spectral dimension, d S , of such a branched polymer phase is 4 3 . This is in good agreement with numerical simulations for large c.

Random walks on combs

“…The upper bound is obtained by a convexity argument. The result (92) is in agreement with d tree s = 4/3 found in [6] by different methods.…”

Statistical ensemble of scale-free random graphs

“…We also list some open questions. In this paper, when necessary, we make use of some results, found in a different context, scattered in earlier publications we have coauthored [15][16][17][18]. We believe, that it is useful to adapt these results to the present context, putting them in a new perspective and making them accessible to a different community.…”

The spectral dimension of random trees

“…These random trees represent a generalization of the incipient infinite percolation clusters on Bethe lattices [5], since the branching probabilities are freely assigned rather than being fixed by a single percolation probability. They also generalize to arbitrary coordination fractions f z the infinitely large specimens of the branched polymers studied for instance in [6].…”

“…for the collection φ = {φ x ; x ∈ B(o, r)} relative to any Van Hove ball (6) is (minus) the Laplacian matrix on G, z x being the coordination or degree of x and A xy the adjacency matrix). The thermodynamic limit is achieved by letting r → ∞ and defines a Gaussian measure for the whole field φ = {φ x ; x ∈ G} which does not depend on the centre o of the ball [13] if G is bounded.…”

“…In this case the definition of spectral dimension itself is subtle since such a concept does not apply to finite graphs. This is the major difficulty in the approach of [6]. Another mathematically well-founded approach to the problem is given in [8], where the exponent θ of the anomalous diffusion on our class of random trees is determined as θ = 1 (the author reports only the shorter and simpler of the two different derivations, which turns out to be more than 20 pages long, and states that the other, more complete treatment is 'monstrously long').…”