@incollection{statphys23_0822,
	title = {The spectral dimension of generic trees},
	address = {Genova, Italy},
	author = {J. Wheater and B. Durhuus and T. Jonsson},
	booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
	editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
	month = {9-13 July},
	url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=822},
	year = {2007},
	biburl = {http://www.bibsonomy.org/bibtex/2e7b9d66e29168fdf0e27383c7e87b95d/statphys23},
	abstract = {We define generic ensembles of infinite trees. These are limits as $N\rightarrow\infty$ 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.  The Hausdorff dimension is defined in terms of $V(R)$, the ensemble average of the volume of a ball of radius $R$, by $V(R)\sim R^{d_h}$. For all these ensembles it is known that $d_h=2$. 

We consider unbiased random walks on these trees. The spectral dimension is given by $p(t)\sim t^{-d_s/2}$ for $t\rightarrow\infty$ where $p(t)$ denotes the return probability for a simple random walk as a function of time. Our main result is a proof that  $d_s=4/3$ for all these ensembles, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is $1/3$.

Our method of proof is conceptually simple. The existence of a unique infinite spine plays a fundamental role because it allows the trees to be decomposed into the (infinite) spine with identically distributed random outgrowths. We then determine $d_s$ by decomposing the walks and obtaining bounds for the leading singular behaviour of the generating function for $p(t)$. These methods will allow the calculation of the spectral dimension of any random tree ensemble with the unique spine property. However, non-generic (sometimes called exotic) trees are not expected to have this property and it would be interesting to see if our methods could be generalized to that case.},
	keywords = {dimension random spectral statphys23 topic-3 trees walk }
}