Curie-Weiss Model under $\ell^{p}$ constraint and a Generalized Hubbard-Stratonovich Transform

We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell^{p}$ norm for some $p>0$. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p>2$, there exists $\beta_{c}(p)$ such that for $\beta<\beta_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. In contrast, for $\beta>\beta_{c}(p)$ the magnetization is concentrated at $\pm m_\ast$ for some $m_\ast>0$. We have $\beta_{c}(p)>1$ for $p>2$ and $\lim_{p\to\infty}\beta_{c}(p)=3$. We further generalize the model for general symmetric spin distributions and prove a similar phase transition. For $0<p<1$, the log-partition function scales at the order of $n^{2/p-1}$. The proofs are based on a generalized Hubbard-Stratonovich (GHS) transform, which is of independent interest.