Abstract
We consider the Berlin–Kac spherical model for supercritical densities under a periodic lattice energy function which has finitely many non-degenerate global minima. Energy functions arising from nearest neighbour interactions on a rectangular lattice have a unique minimum, and in that case the supercritical fraction of the total mass condenses to the ground state of the energy function. We prove that for any sufficiently large lattice size this also happens in the case of multiple global minima, although the precise distribution of the supercritical mass and the structure of the condensate mass fluctuations may depend on the lattice size. However, in all of these cases, one can identify a bounded number of degrees of freedom forming the condensate in such a way that their fluctuations are independent from the rest of the fluid. More precisely, the original Berlin–Kac measure may be replaced by a factorized supercritical measure where the condensate and normal fluid degrees of freedom become independent random variables, and the normal fluid part converges to the critical Gaussian free field. The proof is based on a construction of a suitable coupling between the two measures, proving that their Wasserstein distance is small enough for the error in any finite moment of the field to vanish as the lattice size is increased to infinity.
Highlights
Berlin and Kac proposed [1] in 1952 a spherical model as a modification of the Ising model of a ferromagnet
We prove that for any sufficiently large lattice size this happens in the case of multiple global minima, the precise distribution of the supercritical mass and the structure of the condensate mass fluctuations may depend on the lattice size
The original Berlin–Kac measure may be replaced by a factorized supercritical measure where the condensate and normal fluid degrees of freedom become independent random variables, and the normal fluid part converges to the critical Gaussian free field
Summary
Berlin and Kac proposed [1] in 1952 a spherical model as a modification of the Ising model of a ferromagnet. One of the main goals of the present contribution has been to find methods which would be able to identify the condensate modes properly for general, finite range lattice interactions This has resulted in the bounds given in Theorem 1; as we discuss, these bounds are sufficiently refined to distinguish the condensate modes correctly in the above odd and even L cases, and in all other examples considered in Sect. The fixed finite lattice case for supercritical densities is discussed in Theorem 1 while the conclusions for the case where a given dispersion relation is studied in the infinite volume limit are given in Corollary 1 These results give bounds for the Wasserstein distance between the spherical model measure and the approximation where the condensate and normal fluid modes have been separated. In the two Appendices, we first clarify the precise mathematical interpretation of the δ-function constraints and recall the definition and basic properties of the Wasserstein distance
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
