  We consider a family of continuous time symmetric random walks indexed by
$k\in \mathbb{N}$, $\{X_k(t),\,t\geq 0\}$. For each $k\in \mathbb{N}$ the
matching random walk take values in the finite set of states
$\Gamma_k=\frac{1}{k}(\mathbb{Z}/k\mathbb{Z})$ which is a subset of the unitary
circle. The stationary probability for such process converges to the uniform
distribution on the circle, when $k\to \infty$.
  We disturb the system considering a fixed $C^2$ potential $V: \mathbb{S}^1
\to \mathbb{R}$ and we will denote by $V_k$ the restriction of $V$ to
$\Gamma_k$. Then, we define a non-stochastic semigroup generated by the matrix
$k\,\, L_k + k\,\, V_k$, where $k\,\, L_k $ is the infinifesimal generator of
$\{X_k(t),\,t\geq 0\}$. From the continuous time Perron's Theorem one can
normalized such semigroup, and, then we get another stochastic semigroup which
generates a continuous time Markov Chain taking values on $\Gamma_k$. The
stationary probability vector for such Markov Chain is denoted by $\pi_{k,V}$.
We assume that the maximum of $V$ is attained in a unique point $x_0$ of
$\mathbb{S}^1$, and from this will follow that $\pi_{k,V}\to \delta_{x_0}$. Our
main goal is to analyze the large deviation principle for the family
$\pi_{k,V}$, when $k \to\infty$. The deviation function $I^V$, which is defined
on $ \mathbb{S}^1$, will be obtained from a procedure based on fixed points of
the Lax-Oleinik operator and Aubry-Mather theory.
