The Spectral Dimension of Generic Trees

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 dh=2. Our main result is that the spectral dimension of the ensemble average is ds=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.

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