14/06/2023, 17:00 — 18:00 — Online
Partha Dey, University of Illinois, Urbana-Champaign
Curie-Weiss Model under $l^p$ constraint
We consider the Curie-Weiss model on the complete graph $K_n$ with spin configurations constrained to have a given $l^p$ norm for some $p>0$. For $p=\infty$, this reduces to the classical Ising Curie-Weiss model. We generalize the model with a self-scaled Hamiltonian for general symmetric spin distribution with variance one. Using a modified Hubbard-Stratonovich transform and a coupling of log-gamma distributions, we compute the limiting free energy. As a consequence, we prove that for all $p>1$, there exists a critical $\gb_c(p)$ such that for $\gb<\gb_c(p)$, the magnetization is concentrated at zero and satisfies a Gaussian CLT. In contrast, the magnetization is not concentrated at zero for $\gb>\gb_c(p)$, similar to the classical case. While $\gb_c(2)=1$, we have $\gb_{c}(p)>1$ for $p>2$. To understand the magnetization, we introduce an exchangeable dynamics on the $l^p$ sphere surface, which is of independent interest. For $0 < p < 1$, the log-partition function scales at the order of $n^{2/p-1}$. Based on joint work with Daesung Kim.
See also: https://spmes.impa.br
