  In the present paper, we study the analyticity of the leftmost eigenvalue of
the linear elliptic partial differential operator with random coefficient and
analyze the convergence rate of the quasi-Monte Carlo method for approximation
of the expectation of this quantity. The random coefficient is assumed to be
represented by an affine expansion $a_0(\boldsymbol{x})+\sum_{j\in
\mathbb{N}}y_ja_j(\boldsymbol{x})$, where elements of the parameter vector
$\boldsymbol{y}=(y_j)_{j\in \mathbb{N}}\in U^\infty$ are independent and
identically uniformly distributed on $U:=[-\frac{1}{2},\frac{1}{2}]$. Under the
assumption $ \|\sum_{j\in \mathbb{N}}\rho_j|a_j|\|_{L_\infty(D)} <\infty$ with
some positive sequence $(\rho_j)_{j\in \mathbb{N}}\in \ell_p(\mathbb{N})$ for
$p\in (0,1]$ we show that for any $\boldsymbol{y}\in U^\infty$, the elliptic
partial differential operator has a countably infinite number of eigenvalues
$(\lambda_j(\boldsymbol{y}))_{j\in \mathbb{N}}$ which can be ordered
non-decreasingly. Moreover, the spectral gap
$\lambda_2(\boldsymbol{y})-\lambda_1(\boldsymbol{y})$ is uniformly positive in
$U^\infty$. From this, we prove the holomorphic extension property of
$\lambda_1(\boldsymbol{y})$ to a complex domain in $\mathbb{C}^\infty$ and
estimate mixed derivatives of $\lambda_1(\boldsymbol{y})$ with respect to the
parameters $\boldsymbol{y}$ by using Cauchy's formula for analytic functions.
Based on these bounds we prove the dimension-independent convergence rate of
the quasi-Monte Carlo method to approximate the expectation of
$\lambda_1(\boldsymbol{y})$.
