  The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in
$\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$
where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded
non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an
explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in
terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the
automorphism group of the domain $D_{n,m}(\mu)$. In this article, we obtain
rigidity results on proper holomorphic mappings between two equidimensional
Fock-Bargmann-Hartogs domains. Our rigidity result implies that any proper
holomorphic self-mapping on the Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$
with $m\geq 2$ must be an automorphism.
