Rigidity percolation in a field

Rigidity Percolation with g degrees of freedom per site is analyzed on randomly diluted Erdos-Renyi graphs with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n_f of uncanceled degrees of freedom and gamma_r of redundant bonds. Upon crossing the coexistence line, n_f and gamma_r are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely that the density of uncanceled degrees of freedom is a ``free energy'' for Rigidity Percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1)-core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma_c, Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1)-rigid-core, i.e. the average coordination of these subgraphs is exactly 2g, although gamma_r, the average coordination of the whole system, is smaller than 2g. gamma_c is found to converge to 2g for large g, i.e. in this limit Maxwell counting is exact globally as well. This paper is dedicated to Dietrich Stauffer, on the occasion of his 60th birthday.Comment: RevTeX4, psfig, 16 pages. Equation numbering corrected. Minor typos correcte

Reentry Concerns of Incarcerated Substance Abusers

To facilitate planning of a state-wide reentry program for parolees with histories of drug and alcohol abuse, group interviews were conducted to determine the perceived problems and reentry service needs of 110 male and female inmates of Oregon correctional institutions. Expressed concerns about problems related to frustrations of prison experience, familial and peer relationships, parole supervision, and day-to-day survival skills point to significant gaps and biases in existing reentry services planned from an "objective" viewpoint that slights direct assessment of potential clients' needs