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Let $X$ be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifoldand let $T_P$ be the Toeplitz operator on $X$ associated with a first-orderpseudodifferential operator $P$. In our previous work we established theasymptotic expansion for $k$ large of the kernel of the operators$\chi(k^{-1}T_P)$, where $\chi$ is a smooth cut-off function supported in thepositive real line. By using these asymptotics, we show in this paper that $X$can be projectively embedded by maps with components of the form$\chi(k^{-1}\lambda)f_\lambda$, where $\lambda$ is an eigenvalue of $T_P$ and$f_\lambda$ is a corresponding eigenfunction. We establish the asymptotics ofthe pull-back of the Fubini-Study metric by these maps and we obtain thedistribution of the zero divisors of random Cauchy-Riemann functions. We thenestablish a version of the Lelong-Poincar\'e formula for domains with boundaryand obtain the distribution of the zero divisors of random holomorphicfunctions on strictly pseudoconvex domains.

Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions