The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two multifractals. In this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution. The pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures. The generalized box dimensions of the convolution of Cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization. Returning to the study of this measure we apply the general inequalities and present numerical approximations of the D_q-spectrum. For the first time we are able to obtain results on multifractal properties of a physical quantity in the one-dimensional random-field Ising model which in principle could be measured experimentally. The numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h.

75.10.Hk Classical spin models

75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects

02.50.Cw Probability theory

75.10.Nr Spin-glass and other random models

02.60.-x Numerical approximation and analysis

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) (See also 47B35)

82D40 Magnetic materials

60A10 Probabilistic measure theory (For ergodic theory, see 28Dxx and 60Fxx)

Computational physics

Condensed matter: electrical, magnetic and optical

Issue 39 (5 October 2001)

Received 21 February 2001 , in final form 16 July 2001

Published 21 September 2001