Multiscaling and multifractality in an one-dimensional Ising model

Scaling properties of the Gibbs distribution of a finite-size one-dimensional Ising model are investigated as the thermodynamic limit is approached. It is shown that, for each nonzero temperature, coarse-grained probabilities of the appearance of particular energy levels display multiscaling with the scaling length ℓ = 1/Mn, where n denotes the number of spins and Mn is the total number of energy levels. Using the multifractal formalism, the probabilities are argued to reveal also multifractal properties.