\lesssim
\left( \left\vert \alpha \right\vert +\beta \right) ^{n-1}, \\
\left\vert \partial _{\overline{\lambda }}G\right\vert & \lesssim
_{q_{+}}\exp \left\{ -Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}\right\}
.\end{aligned}$$ Thus $$\left\vert F(\alpha ,\beta )\right\vert \lesssim _{q_{+}}e^{\varkappa \left(
\varkappa ^{2}t+\left\vert z\right\vert \right) }\left( \left\vert \alpha
\right\vert +\beta \right) ^{n-1}e^{-\left( \sqrt{24\beta t}\alpha +\frac{{\operatorname{Im}}z}{\sqrt{24\beta t}}\right) ^{2}}\exp \left\{ \frac{{\operatorname{Im}}^{2}z}{24t}\beta
^{-1}-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}\right\} .
\label{eq5.4.0}$$ To prove Theorem \[thm1.2\] we only need to consider the case $n=1$. We have $$\begin{gathered}
  \label{eq5.4.1}
\int_{0}^{\varkappa /2}\frac{d\beta }{\beta }\int_{-\infty }^{\infty
}\left\vert F\left( \alpha ,\beta \right) \right\vert d\alpha \\
\lesssim _{z,t,q_{+}}\int_{0}^{\varkappa /2}\beta ^{-3/2}\exp \left\{ \frac{{\operatorname{Im}}^{2}z}{24t}\beta ^{-1}-Q\beta ^{-\frac{\varepsilon }{1-\varepsilon }}\right\} d\beta .\end{gathered}$$ So $F$ is subject to the condition of Lemma \[lem5.1\] if the integral in converges, which depends on $\varepsilon$ and ${\operatorname{Im}}z$.

-   $\varepsilon >1/2$. Then[^7] $\frac{\varepsilon }{1-\varepsilon }>1$ and the right hand side of is finite for any $z\in \mathbb{C}$. This means that $\mathbb{M}_{+}(z,t)$ is an entire $\mathfrak{S}_{1}$-valued function for any $t>0$ and due to , we can pass to the limit in as $n\rightarrow \infty $ by standard properties of infinite determinants (see e.g. [@GGK00]). This proves in Theorem \[thm1.2\].

-   $\varepsilon =1/2$. Then $\frac{\varepsilon }{1-\varepsilon }=1$ and the right hand side of converges if and only if $$\frac{{\operatorname{Im}}^{2}z}{24t}-Q<0$$or when $$\left\vert {\operatorname{Im}}z\right\vert <\sqrt{12Q}\cdot \sqrt{t}.$$Choosing 