content_image/338582.jpg content_image/338583.jpg content_image/338579.jpg content_image/338580.jpg content_image/338578.jpg content_image/338576.jpg content_image/338577.jpg
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). 

## B. The general case 

<img src='content_image/338582.jpg'>

<img src='content_image/338583.jpg'>

<img src='content_image/338579.jpg'>

<img src='content_image/338580.jpg'>

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 

<img src='content_image/338578.jpg'>

<img src='content_image/338576.jpg'>

<img src='content_image/338577.jpg'>

