Spatial fluctuations of the local electric field induced by a constant applied electric field in composite media are studied analytically and numerically. It is found that the density of states for the fields exhibit sharp peaks and abrupt changes in the slope at certain critical points which are analogous to van Hove singularities in the density of states for phonons and electrons in solids. As in solids, these singularities are generally related to saddle and inflection points in the field spectra and are useful in characterizing field fluctuations. The critical points are very prominent in dispersions with a regular, ‘‘crystal-like,’’ structure. However, they broaden and eventually disappear as the disorder increases. @S0163-1829~98!50542-5# In the study of heterogeneous materials, the preponderance of work has been devoted to finding the effective transport, electromechanical, and mechanical properties of the material, 1 which amounts to knowing only the first moment of the local field. When composites are subjected to constant applied fields, the associated local fields exhibit strong spatial fluctuations. The analysis and evaluation of the distribution of the local field has received far less attention. Nonetheless, the distribution of the local field is of great fundamental and practical importance in understanding many crucial material properties such as breakdown phenomenon 2 and the nonlinear behavior of composites. 3 Much of the work on field distributions has been carried out for lattice models using numerical 4,5 and perturbation methods. 6 Recently, continuum models have been also addressed using numerical techniques. 7 In this paper, we study the local electric field fluctuations by analyzing the density of states for the fields. To illustrate the procedure, we evaluate the density of states for three different continuum models of dielectric composites: the Hashin-Shtrikman ~HS! construction, 8 periodic, and random arrays of cylinders. It is found that the density of states for the fields exhibits sharp peaks and abrupt changes in the slope at certain critical points which are analogous to van Hove singularities in the density of states for phonons and electrons in solids. This analogy is useful in quantifying field fluctuations in composites. In the case of the HashinShtrikman construction, we obtain an exact analytical expression for the density of states. We first describe the basic equations and then determine the density of states for the aforementioned examples. Consider a composite material composed of (n21) isotropic inclusions with dielectric constants e i and volume fractions f i (i52,...,n) in a uniform reference matrix of dielectric constant e 1 with volume fraction f 1 . Clearly, the local dielectric constant at position r is e(r) 5( i51