2025/09/11 by Boursier, Jeanne, Serfaty, Sylvia · 1 citation
#82B05 #82B26 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2509.09449
We study the two-dimensional two-component Coulomb gas in the canonical ensemble and at inverse temperature β>2. In that regime, the partition function diverges and the interaction needs to be cut off at a length scale λ>0. Particles of opposite charges tend to pair into dipoles of length scale comparable to λ, which themselves can aggregate into multipoles. Despite the slow decay of dipole--dipole interactions, we construct a convergent cluster expansion around a hierarchical reference model that retains only intra-multipole interactions. This yields a large deviation result for the number of 2p-poles as well as a sharp free energy expansion as N→∞ and λ→0 with three contributions: (i) the free energy of N independent dipoles, (ii) a perturbative correction, and (iii) the contribution of a non-dilute subsystem. The perturbative term has two equivalent characterizations: (a) a convergent Mayer series obtained by expanding around an i.i.d. dipole model; and (b) a variational formula as the minimum of a large-deviation rate function for the empirical counts of 2p-poles. The Mayer coefficients exhibit transitions at βp=4-\tfrac2p, that accumulate at β=4, which corresponds to the Berezinskii-Kosterlitz-Thouless transition in the low-dipole-density limit. At β=βp the p-dipole cluster integrals switch from non-integrable to integrable tails. The non-dilute system corresponds to the contribution of large dipoles: we exhibit a new critical length scale Rβ, λ which transitions from λ-(β-2)/(4-β) to +∞ as β crosses the critical inverse temperature β=4, and which can be interpreted as the maximal scale such that the dipoles of that scale form a dilute set.