We study the system formed by a gaz of black holes and strings within a microcanonical formulation .
The density of mass levels grows asymptotically as \rho ( m ) \approx ( d m _ { i } + bm _ { i } ^ { 2 } ) ^ { - a } e ^ { \frac { 8 \pi } { 2 } ( d m _ { i } + bm _ { i } ^ { 2 % } ) } , ( i = 1 , ... ,N ) .
We derive the microcanonical content of the system : entropy , equation of state , number of components N , temperature T and specific heat .
The pressure and the specific heat are negative reflecting the gravitational unstability and a non-homogeneous configuration .
The asymptotic behaviour of the temperature for large masses emerges as the Hawking temperature of the system ( classical or semiclassical phase ) in which the classical black hole behaviour dominates , while for small masses ( quantum black hole or string behavior ) the temperature becomes the string temperature which emerges as the critical temperature of the system .
At low masses , a phase transition takes place showing the passage from the classical ( black hole ) to quantum ( string ) behaviour .
Within a microcanonical field theory formulation , the propagator describing the string-particle-black hole system is derived and from it the interacting four point scattering amplitude of the system is obtained .
For high masses it behaves asymptotically as the degeneracy of states \rho ( m ) of the system ( ie duality or crossing symmetry ) .
The microcanonical propagator and partition function are derived from a ( Nambu-Goto ) formulation of the N-extended objects and the mass spectrum of the black-hole-string system is obtained : for small masses ( quantum behaviour ) these yield the usual pure string scattering amplitude and string-particle spectrum M _ { n } \approx \sqrt { n } ; for growing mass it pass for all the intermediate states up to the pure black hole behaviour .
The different black hole behaviours according to the different mass ranges : classical , semiclassical and quantum or string behaviours are present in the model .