Let $S = (S_t, t \geq 0)$ be a simple one-dimensional continuous-time random walk with total jump rate one, $S_0 = 0$. Denote by $T_k$ the time when $S$ exits the interval $I_k = [-k,k] \cap \mathbb{Z}$. Let also for an interval of integers $I$, $\lambda (I)$ be the principle Dirichlet eigenvalue of the normalized discrete Laplacian on $I$ defined as $\Delta f(y) = \frac 12 [f(x+1) + f(x-1) - 2f(x) ]$ for $f$ vanishing outside $I$; that is, $\lambda(I) = 1 - \cos (\frac{\pi}{\ell + 1})$, where $\ell$ is the length of $I$.

In [1] in Section 5 the following inequality is used

$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P (T_k > q) \leq c (\lambda (I_k) q + 1 ) ^{1/2} e^{- \lambda (I_k) q}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$ where $c >0 $ is a constant. The authors say that (1) can be obtained "from standard estimates" and give two references. One of the references is not accessible to me, while I have not found the other very helpful.

I would thus appreciate any reference where inequalities of type (1) involving the principle eigenvalue of the Laplacian are discussed.

[1] *Ramírez, A. F.; Sidoravicius, V.*, **Asymptotic behavior of a stochastic combustion growth process**, J. Eur. Math. Soc. (JEMS) 6, No. 3, 293-334 (2004). ZBL1049.60089.