On a third-order phase transition

The asymptotic behaviour of random variables of the general form $$\ln \sum\limits_{i = 1}^{\kappa ^N } {\exp (N^{1/p} \beta \zeta _i )} $$ with independent identically distributed random variables ζ _i is studied. This generalizes the random energy model of Derrida. In the limitN→∞, there occurs a particular kind of phase transition, which does not incorporate a bifurcation phenomenon or symmetry breaking. The hypergeometric character of the problem (see definitions of Sect. 4), its Φ-function, and its entropy function are discussed.