复数空间中具有指数分布的随机P多项式的零点
Zeros of random $P$-polynomials in $\mathbb{C}^d$ with exponential profiles
AI总结：
研究复数空间中具有指数分布的随机多变量P多项式的零点，证明归一化势函数收敛到确定性托里克 plurisubharmonic 函数，并讨论零点流的弱收敛性。
摘要：
我们研究具有单调支撑的随机多变量P多项式，其系数具有均匀指数分布。在尾部条件假设下，证明归一化势函数在局部L^1意义下几乎处处收敛到确定性托里克 plurisubharmonic 函数Φ_{P,f}，并由此得出零点流的弱收敛性。在更强的对数矩假设下，证明零点流在d>2时几乎处处收敛，d≤2时沿稀疏子序列收敛。在(ℂ^*)^d上，极限势函数由Φ_{P,f}(z)=I_{P,f}(Log z)给出，其中I_{P,f}是P上分布的Legendre-Fenchel变换，Log z=(log|z_1|,...,log|z_d|)。这些结果将Kabluchko和Zaporozhets的指数分布机制从单复变扩展到真正的多变量P多项式设置，连接了随机零超曲面与由(P,f)确定的凸分析数据。
英文摘要：
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 $\mathbb{P}(\log(1+|ξ_0|)>t)=o(t^{-d})$ 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 $Φ_{P,f}$, and consequently the normalized zero currents $\frac1n[Z_{\mathbf{P}_n}]$ converge weakly to the closed positive $(1,1)$-current $dd^cΦ_{P,f}$. Under the stronger logarithmic moment assumption $\mathbb{E}[(\log(1+|ξ_0|))^d]<\infty$, 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 $Φ_{P,f}(z)=I_{P,f}(\operatorname{Log} z)$, 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 $(P,f)$.
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