Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix $$\Delta_D := \left( \begin{matrix} 2 & -1 & \\ -1 & 2 & -1 \\ & -1 & 2 & -1 \\ & & -1 &2 \end{matrix}\right)\in \mathbb C^{4 \times 4}.$$

This matrix is self-adjoint. It has an eigendecomposition with eigenvectors $(v_i).$

In particular, we can decompose the first unit vector in its eigenbasis $$1 = \sum_{i=1}^n \lvert\langle v_i^{n},e_1 \rangle \rvert^2.$$

Clearly, as $n$ tends to infinity the convergence of the series implies that the smallest object $\inf_i\lvert\langle v_i^{n},e_1 \rangle \rvert^2$ decays faster than $1/n.$

I would like to know: Can one find the asymptotics of $$\inf_i\lvert\langle v_i^{n},e_1 \rangle \rvert^2$$