Let S n = X 1 + • • • + X n be an irreducible random walk on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments X n and obtain an upper and lower bounds of the potential function, a(x), of S n in the form a(x) ≍ x/m(x) under a reasonable condition on the distribution of X n ; we especially show that as x → ∞