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**](http://dx.doi.org/10.4171/JEMS/11), J. Eur. Math. Soc. (JEMS) 6, No. 3, 293-334 (2004). [ZBL1049.60089](https://zbmath.org/?q=an:1049.60089).



Exit time estimate for a simple continuous-time random walk
Let $\tau\_k$ denote the number of steps for discrete time simple RW on the integers (started at 0) to exit the interval $[-k,k]$. Let $\gamma:=\cos(\frac{\pi}{2k+2})$ so that $\lambda=1-\gamma$.
The formula in [1] page 243, line -5, gives (bounding the alternating series there by twice the first term and using
$\cot(x) \le 1/\sin(x) \le \pi/(2x)$ for $x\in[0,\pi/2]$ ) that
$$P[\tau\_k>n] \le 8\gamma^n \,.$$
The estimate for continuous time RW follows:
$$P[T\_k>q] \le \sum\_n P[{\rm Poisson}(q)=n] \cdot P[\tau\_k>n] \,$$
whence
$$P[T\_k>q] \le \sum\_n \frac{q^n e^{-q}}{n!}\cdot 8\gamma^n =8e^{q\gamma-q}=8e^{-q\lambda}\,.$$


[1] Spitzer, Frank. Principles of random walk. GTM Vol. 34. Second edition, Springer.


