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Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls

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Fabio Deelan Cunden, Paolo Facchi, Pierpaolo Vivo, Marilena Ligabo'

Original languageEnglish
Pages (from-to)1262-1297
Number of pages36
JournalJournal of Statistical Physics
Issue number6
Early online date11 Apr 2019
Accepted/In press28 Mar 2019
E-pub ahead of print11 Apr 2019
Published30 Jun 2019


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Consider the free energy of a d-dimensional gas in canonical equilibriumunder pairwise repulsive interaction and global confinement, in presence of a volumeconstraint. When the volume of the gas is forced away from its typical value, thesystem undergoes a phase transition of the third order separating two phases (pulledand pushed).We prove this result i) for the eigenvalues of one-cut, off-critical randommatrices (log-gas in dimension d=1) with hard walls; ii) in arbitrary dimension d>=1for a gas with Yukawa interaction (aka screened Coulomb gas) in a generic confiningpotential. The latter class includes systems with Coulomb (long range) and delta (zerorange) repulsion as limiting cases. In both cases, we obtain an exact formula forthe free energy of the constrained gas which explicitly exhibits a jump in the thirdderivative, and we identify the ‘electrostatic pressure’ as the order parameter of thetransition. Part of these results were announced in [F. D. Cunden, P. Facchi, M. Ligabo'and P. Vivo, J. Phys. A: Math. Theor. 51, 35LT01 (2018)].

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