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

We consider the Ising Curie-Weiss model on the complete graph constrained under a given ${\mathrm{\ell }}^p$ norm for some $p>0$. For $p=\mathrm{\infty }$, 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 ${\lim }_{p\to \mathrm{\infty }}{\beta }_c(p)=3$. 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 $n^{2\mathrm{/}p-1}$. The proofs are based on a generalized Hubbard-Stratonovich (GHS) transform, which is of independent interest.