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Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the onedimensional integer lattice with zero mean, infinite variance and i.i.d.increments $X_n$. We obtain an upper and lower bounds of the potentialfunction, $a(x)$, of $S_n$ in the form $a(x)\asymp x/m(x)$ under a reasonablecondition on the distribution of $X_n$; we especially show that as $x\to\infty$$$a(x) \asymp \frac{x}{m_-(x)} \quad\mbox{and}\quad \frac{a(-x)}{a(x)} \to 0\quad\;\;\mbox{if}\quad \lim_{x\to +\infty} \frac{m_+(x)}{m_-(x)} =0,$$ where$m_\pm(x) = \int_0^xdy\int_y^\infty P[\pm X_1>u]du$ and $m=m_++m_-$. Undercertain conditions on the tails of the distribution of $X$ we derive preciseasymptotic forms of $a(x)$ as $x\to +\infty$ or/and $-\infty$. The results areapplied to derive a sufficient condition for the relative stability of theladder height and estimates of some escape probabilities from the origin; weshow among others that under the above condition on $m_+/m-$, $P[S_n>0] \to1/\alpha$ if and only if the probability of exiting a long interval $[-Q,R]$through the upper boundary converges to $\lambda^{\alpha-1}$ as $Q/(Q+R) \to\lambda$ for any $0<\lambda<1$.

Estimates of Potential functions of random walks on $Z$ with zero mean and infinite variance and their applications