Large deviations for random walk in random environment with cookies

New results will be presented about asymptotics of large deviations probabilities for random walk in random environment with cookies (RWREwC). Let us denote the process. Let
$$ \vec{p} = \{p(i)\}_{i \in \mathbb{Z}}, \quad \vec{p_1} = \{p_1(i)\}_{i \in \mathbb{Z}} $$
be independent sequences of independent identically distributed random variables taking values in $(0, 1)$. Let a sequence $\{X_n\}_{n \in \mathbb{N}}$ be denoted as follows:
$$ \mathbf{P}(X_{n+1} = 1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) = 0) = p_1(i), \\ \mathbf{P}(X_{n+1} = -1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) = 0) = 1 - p_1(i), \\ \mathbf{P}(X_{n+1} = 1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) > 0) = p_1(i), \\ \mathbf{P}(X_{n+1} = -1|\vec{p}, \vec{p_1}, X_1, . . . , X_n, S_n = i, D_n(i) > 0) = 1 - p_1(i). $$
Here
$$ S_0 := 0,\: S_n := \sum_{i=1}^{n}X_i,\quad D_n(i) := \sum_{j=1}^{n}I(S_{j-1} = i, X_j = -1),\quad n \in \mathbb{N},\: i \in \mathbb{Z}. $$
The process $\{S_n\}_{n \ge 0}$ is called random walk in random environment with cookies. It generalizes the classical model of random walk in random environment (RWRE) inroduced by F. Solomon in [1]. If the laws of distribution of $p_1(0)$ and $p(0)$ are the same, then the models of RWRE and RWREwC are coincide. Limit theorems for RWRE were obtained in [2]. Precise asymptotics of large deviations probabilities for the first moment RWRE reaching level $n$ were obtained in [3] by the author.
By definition, put
$$ T_0 := 0,\quad T_n := \min\{k \in \mathbb{N} : S_k = n\},\quad n \in \mathbb{N}. $$
The results on precise asymptotics of $\mathbf{P}(T_n = k)$ in the RWREwC model will be presented.

References

1. Solomon F., “Random walks in a random environment”, The Annals of Probability, 3:1 (1975), 1–31
2. Kesten H., Kozlov M., Spitzer F., “A limit law for random walk in a random environment”, Compositio mathematica, 30:2 (1975), 145–168
3. Bakai G. A., “O bolshikh ukloneniyakh momenta dostizheniya dalekogo urovnya sluchainym bluzhdaniem v sluchainoi srede”, Diskretnaya matematika, 34:4 (2022), 3–13