we get namely with δ ± j ≡ λ 2 $_{ν}$H j ± 1 j . It is worth commenting here that this last expression was also pointed out in [9], discussing aspects of discretization instabilities and the relation to the directed polymer problem. We will further discuss the mapping to the directed polymer problem in Sec. II C below. By expanding the exponentials up to terms of order of a $^{2}$, and collecting equal powers of a (observe that the zero-order contribution vanishes) we retrieve with ( Q stands for “quadratic”). As we see, the first and second terms on the r.h.s. of Eq. (7) are necessarily related by virtue of Eq. (6). A Taylor expansion of φ$_{j}$$_{+}$$_{l}$ around φ$_{j}$ shows that the general form of the discrete Lapla- cian, involving up to the n –th nearest neighbors of site j , is of the form where as before, the subscript stands for the number of nearest neighbors. Since the maxi- mum value for n is M ≡ ( N − 1) \ 2, where \ denotes integer division, one may alternatively run the sum up to M and set b$_{l}$ = 0 , l = n + 1 . . . M . The remaining b$_{l}$ , that are other- wise arbitrary, should be fixed by whatever criterion (below, we shall use the criterion of maximizing accuracy). Repeating the steps described above, one obtains 6 L$_{(}$$_{n}$$_{)}$ ( h$_{j}$ ) = ∑ n l $_{=1}$b$_{l}$ [ H j + l j + H j − l j ] a ∑ n l $_{=1}$l$^{2}$b$_{l}$ , (10) n j + l 2 j − l 2 Q$_{(}$$_{n}$$_{)}$ ( h$_{j}$ ) = ∑ n l $_{=1}$b$_{l}$ [ ( H j + l j ) 2 + ( H j − l j ) 2 ] 2 ∑ n l $_{=1}$l$^{2}$b$_{l}$ . (11) L$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ) = ∑ n l $_{=1}$b$_{l}$ [ Φ j + l j + Φ j − l j ] a ∑ n l $_{=1}$l$^{2}$b$_{l}$ , (9) ˙ h$_{j}$ = ν L$_{(1)}$ ( h$_{j}$ ) + λ 2 Q$_{(1)}$ ( h$_{j}$ ) + ε ξ$_{j}$, (7) Q$_{(1)}$ ( h$_{j}$ ) = 1 2 [ ( H j +1 j ) 2 + ( H j − 1 j ) 2 ] , (8) B. The general case e λ 2 ν h$_{j}$ λ 2 ν ˙ h$_{j}$ = ν a $_{2}$L$_{(1)}$ ( e λ 2 ν h$_{j}$ ) + λε 2 ν e λ 2 ν $^{h$_{j}$}$ξ$_{j}$ , ˙ h$_{j}$ = 2 ν 2 λa 2 [ e δ + j a + e δ − j a − 2 ] + ε ξ$_{j}$,