content_image/367169.jpg content_image/367168.jpg content_image/367167.jpg content_image/367166.jpg content_image/367165.jpg
There is no loss of generality in taking j = N ≡ 0. If we rearrange the sum in Eq. (21) as ∑ M j = M +1 − $_{N}$, with M ≡ ( N − 1) \ 2, then only − n ≤ j ≤ n will contribute in Eq. (22). Moreover, their contribution is such that they cancel the factor 1/2 in front of the sum in Eq. (21). For completeness, let us show the particular functional form for Q$_{(1)}$ ( φ$_{j}$ ) 

## A. Other discrete forms of the KPZ term 

Of course, Eq. (22) does not uniquely determine the Lyapunov function. Expressions other than Eq. (21) may yield L$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ), provided that they contain the right terms, in the right proportion. Take as an example the proposal of Refs. [10, 11], coded as N (1 / 2) j in Ref. [19] 

By using N (1 / $^{2)}$( φ$_{j}$ ′ ) instead of Q$_{(}$$_{n}$$_{)}$ ( φ$_{j}$ ′ ) in Eq. (20), we obtain 

<img src='content_image/367169.jpg'>

<img src='content_image/367168.jpg'>

<img src='content_image/367167.jpg'>

which is an instance of L$_{(2)}$ ( φ$_{j}$ ), with b$_{1}$ = 2, b$_{2}$ = 1. As it was shown before, the procedure outlined in Sec. II will yield L (1 / 2) (2) ( h$_{j}$ ), together with 

<img src='content_image/367166.jpg'>

and not N (1 / $^{2)}$( h$_{j}$ ). Hence, the proposal of Refs. [10, 11] is not consistent: On one hand, N (1 / $^{2)}$( h$_{j}$ ) does not correspond with L$_{(1)}$ ( h$_{j}$ ), as it is used. On the other hand, it does not correspond with L (1 / 2) (2) ( h$_{j}$ ) either, as shown. 

As stated before, the proposal of Refs. [10, 11] belongs to a family coded in Ref. [19] as N ( γ ) j , γ ∈ [0 , 1]. The choice γ = 1 yields N $^{(1)}$( φ$_{j}$ ) = 1 4 ( Φ j +1 j − 1 ) $^{2}$. On the other hand, the choice γ = 0 yields L (0) $_{(2)}$( h$_{j}$ ) = L$_{(1)}$ ( h$_{j}$ ). They all correspond to n = 2, with b$_{1}$ = 1 − γ , b$_{2}$ = γ/ 2. The equivalent of Eq. (24) is now 

<img src='content_image/367165.jpg'>

