Exit time estimate for a simple continuous-time random walk

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.

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 $$P[T_k>q] \le \sum_n \frac{q^n e^{-q}}{n!}\cdot 8\gamma^n =8e^{q\gamma-q}=8e^{-q\lambda}\,.$$