Emergence of Concentration Effects in the Variational Analysis of the N‐Clock Model. Issue 10 (22nd December 2021)

• Record Type: Journal Article
• Authors: Cicalese, Marco; Orlando, Gianluca; Ruf, Matthias
• Is Part Of: Communications on pure and applied mathematics. Volume 75:Issue 10(2022)
• Page Start: 2279
• Page End: 2342
• Publication Date: 2021-12-22
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.22033
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract: We investigate the relationship between the N ‐clock model (also known as planar Potts model or ℤ N ‐model) and the XY model (at zero temperature) through a Γ‐convergence analysis of a suitable rescaling of the energy as both the number of particles and N diverge. We prove the existence of rates of divergence of N for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the N ‐clock model in this regime features an energy that may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex‐vortex interaction energy. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.