Zeros of random $P$-polynomials in $\mathbb{C}^d$ with exponential profiles

Abstract

We study random multivariate $P$-polynomials in $\mathbb{C}^d$ with monomial supports constrained to $nP\cap\mathbb{Z}_+^d$ for a convex body $P\subset\mathbb{R}_+^d$, and deterministic coefficients admitting a uniform exponential profile $f$ on $P$. Assuming the tail condition $\mathbb{P}(\log(1+|\xi_0|)>t)=o(t^{-d})$ on the i.i.d. complex coefficients, we prove that the normalized potentials $\frac1n\log|\mathbf{P}_n|$ converge in probability in $L^1_{\mathrm{loc}}(\mathbb{C}^d)$ to a deterministic toric plurisubharmonic function $\Phi_{P,f}$, and consequently the normalized zero currents $\frac1n[Z_{\mathbf{P}_n}]$ converge weakly to the closed positive $(1,1)$-current $dd^c\Phi_{P,f}$. Under the stronger logarithmic moment assumption $\mathbb{E}[(\log(1+|\xi_0|))^d]<\infty$, we prove almost sure weak convergence of the zero currents along the full sequence for $d>2$, and along sparse subsequences for $d \le 2$. On $(\mathbb{C}^*)^d$, the limiting potential is given by $\Phi_{P,f}(z)=I_{P,f}(\operatorname{Log} z)$, where $I_{P,f}$ is the Legendre-Fenchel transform of the profile over $P$ and $\operatorname{Log} (z)=(\log|z_1|,\dots,\log|z_d|)$. These results extend the exponential-profile mechanism of Kabluchko and Zaporozhets from one complex variable to the genuinely multivariate $P$-polynomial setting under relaxed probabilistic assumptions, directly connecting random zero hypersurfaces with convex-analytic data determined by $(P,f)$.