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Estimation of critical exponents from the cluster coefficients: Application to hard spheres

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal symmetric matrix, whose elements converge to a constant with a 1/n^2 correction. We find exact expressions, in terms of these correction terms, for the two critical exponents describing the density near the two singular termination points of the fluid phase. We apply the method to the hard-spheres model and find that the metastable fluid phase terminates at rho_t=0.751(5). The density near the transition is given by (rho_t-rho)~(z_t-z)^sigma', where the critical exponent is predicted to be sigma'=0.0877(25). The termination density is close to the observed glass transition, and thus the above critical behavior is expected to characterize the onset of glassy behavior in hard spheres.