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In the present paper, we study the analyticity of the leftmost eigenvalue ofthe linear elliptic partial differential operator with random coefficient andanalyze the convergence rate of the quasi-Monte Carlo method for approximationof the expectation of this quantity. The random coefficient is assumed to berepresented 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 andidentically uniformly distributed on $U:=[-\frac{1}{2},\frac{1}{2}]$. Under theassumption $ \|\sum_{j\in \mathbb{N}}\rho_j|a_j|\|_{L_\infty(D)} <\infty$ withsome 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 ellipticpartial differential operator has a countably infinite number of eigenvalues$(\lambda_j(\boldsymbol{y}))_{j\in \mathbb{N}}$ which can be orderednon-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 $\mathbb{C}^\infty$ andestimate mixed derivatives of $\lambda_1(\boldsymbol{y})$ with respect to theparameters $\boldsymbol{y}$ by using Cauchy's formula for analytic functions.Based on these bounds we prove the dimension-independent convergence rate ofthe quasi-Monte Carlo method to approximate the expectation of$\lambda_1(\boldsymbol{y})$.

Analyticity of Parametric Elliptic Eigenvalue Problems and Applications to Quasi-Monte Carlo Methods