Quenched thermodynamic states of the Ising model on random graphs

Quenched thermodynamic states of the Ising model on random graphs

Yuri Kozitsky

Institute of Mathematics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland

March 1, 2011

Abstract

It is proven that the zero external field Ising ferromagnet on a scale- free graph withpk ∼k^−λ andλ ≤3 can be in a paramagnetic state at finite temperatures, which contradicts the common belief that this prop- erty holds only ifP

kpkk²<∞. Namely, we prove this result for the ge- nealogic tree of a non-extinct Galton-Watson process withP

kpkklnk <

∞. The proof consists in showing that the magnetizationM vanishes if 2βJ <ln(a/a−1), wherea=P

kkpk. It is based on the Kesten-Stigum theorem and on a special representation ofM.

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