Assuming that $({X_n})_{n\in\Bbb{N}}$ describe a simple random walk on the Torus $T_N^d$ and that $Y_t={X_S}_t$ is a lazy random walk (with $S_t=\sum_{i=1}^t\gamma_i$ where $(\gamma_i)_{i\ge1}$ is B(1,$\frac12$) ),
show that for any $\epsilon>0$ and $\delta>0$, there exist C=C($\epsilon,\delta$) and $\beta$=$\beta(\epsilon)\in(0,1)$ such that for all $N\ge1$ (size of $T_N^d$), $K\subset\subset T_N^d$, and $n=\lfloor N^\delta\rfloor$,
$(1-C*\beta^n)*P[\{Y_0,...,Y_{2(1-\epsilon)n}\} \bigcap \not= \emptyset]\le P[\{X_0,...,X_n\} \cap K\not= \emptyset]\le (1+C*\beta^n)*P[\{Y_0,...,Y_{2(1+\epsilon)n}\}\cap K \not= \emptyset ]$
By using the following Lemma: for any $\epsilon>0$ there exists $\alpha=\alpha(\epsilon)\in(0,1)$ such that for all $n\ge 0$
$P_x[\{Y_0,...,Y_{2(1-\epsilon)n}\} \subset \{X_0,...,X_n\}\subset \{Y_0,...,Y_{2(1+\epsilon)n}\}]\ge 1-2*\alpha^n$
one can get the following: for all $x\in K$
$P[\{Y_0,...,Y_{2()1+\epsilon)n}\}\cap K \not= \emptyset]-2*\alpha^n \le P[\{X_0,...X_n\}\cap K \not= \emptyset] \le P[\{Y_0,...Y_{2(1+\epsilon)n}\}\cap K \not= \emptyset]+2*\alpha^n$
After this result it would, in my opinion, suffice to show that,
$(1-C*\beta^n)*P[\{Y_0,...,Y_{2(1-\epsilon)n}\} \bigcap \not= \emptyset]\le P[\{Y_0,...,Y_{2()1+\epsilon)n}\}\cap K \not= \emptyset]-2*\alpha^n $ and $(1+C*\beta^n)*P[\{Y_0,...,Y_{2(1+\epsilon)n}\}\cap K \not= \emptyset] \ge P[\{Y_0,...Y_{2(1+\epsilon)n}\}\cap K \not= \emptyset]+2*\alpha^n$
After using the Lemma and getting the first inequality my book says the following about the problem:
Since $X_0$ has uniform distribution over the vertices of $T_N^d$ over P, we can say that $P[\{X_0,...X_n\}\cap K \not= \emptyset] \ge > P[X_0 \in K] \ge \frac1{N^d}$ We take $\beta=\alpha^\frac12$. Let $N_1=N_1(\epsilon,\delta)$ be such that for all $N \ge N_1$, one has $2*\alpha^n*N^d<1$. For any $N \le N_1$ the inequalities hold for some $C=C(\epsilon,\delta,N_1)$ (this part I understood). For any $N_1>N$ the inequalities hold for $C=C(\epsilon,\delta)$ such that $(1-C\beta^n)*(1+2\alpha^nN^d)\le 1$ and $(1+C\beta^n)*(1-2\alpha^nN^d)\ge 1$
I simply don't get how the last two inequalities imply that the first inequality(the one I need to show) is true for all $N>N_1$.