The spherical model on graphs

We extend the lattice spherical model of Berlin and Kac to infinite graphs (describing inhomogeneous structures such as fractals, polymers and amorpho us materials). We analytically calculate the exact values of the critical e xponents, which turn out to depend only on the vibrational spectral dimensi on (d) over bar of the graph. This functional dependence coincides with the analytic continuation in d of the corresponding exponents for the lattice model. This result provides an example of geometrical universality classes for non-translationally invariant systems and strongly suggests considering (d) over bar as the natural generalization of the Euclidean dimension d fo r critical phenomena.