C. Few remarks on the directed polymer problem We devote this subsection to briefly comment about the mapping of KPZ onto the directed polymer problem. Such a mapping can be carried out via the Hopf-Cole transformation [9] and the resulting linear equation corresponds to Eq. (2). In order to employ the usual rules of calculus, here we assume the Stratonovich interpretation for the multiplicative noise. The corresponding finite differences scheme is, explicitly, As indicated in Eq. (5), the discrete noise ξ$_{j}$ ( t ) is a Gaussian random variable. The mean value of Eq. (12) is One immediately realizes that the drift of this equation becomes singular in the continuum limit a → 0, so one has to renormalize this theory [29–31]. This is done by decomposing the bare parameter into an effective and a singular component, F = F$_{eff}$ + F$_{s}$ , with F$_{s}$ = − 1 / (2 a ). The resulting equation is then which is finite, but in which F$_{eff}$ has to be measured directly from the experiment. Thus the correct interpretation of the Stratonovich Eq. (12) is the following Itˆ o equation In order to measure the effective growth rate, one can solve the linear Eq. (14) to find the globally stable solution 〈 φ$_{j}$ ( t ) 〉 = 〈 φ$_{j}$ (0) 〉 exp[ λF$_{eff}$ t/ (2 ν )] for a spatially homogeneous initial condition 〈 φ$_{j}$ (0) 〉 = 〈 φ (0) 〉 . And so this effective rate can be measured from experi- mental/numerical data in the following fashion or alternatively F$_{eff}$ = 2 ν λt ln {〈 exp[ λh$_{j}$ ( t ) / (2 ν )] 〉} , (17) F$_{eff}$ = 2 ν λt ln[ 〈 φ$_{j}$ ( t ) 〉 / 〈 φ$_{j}$ (0) 〉 ] , (16) ˙ φ$_{j}$ = ν φ$_{j}$$_{+1}$ + φ$_{j}$ − 1 − 2 φ$_{j}$ a 2 + λF$_{eff}$ 2 ν φ$_{j}$ + λε 2 ν φ$_{j}$ξ$_{j}$ ( t ) . (15) d 〈 φ$_{j}$ 〉 dt = ν 〈 φ$_{j}$$_{+1}$ 〉 + 〈 φ$_{j}$ − $_{1}$〉 − 2 〈 φ$_{j}$ 〉 a 2 + λF$_{eff}$ 2 ν 〈 φ$_{j}$ 〉 , (14) d 〈 φ$_{j}$ 〉 dt = ν 〈 φ$_{j}$$_{+1}$ 〉 + 〈 φ$_{j}$ − $_{1}$〉 − 2 〈 φ$_{j}$ 〉 a 2 + λF 2 ν 〈 φ$_{j}$ 〉 + λε 4 νa 〈 φ$_{j}$ 〉 . (13) ˙ φ$_{j}$ = ν L$_{(1)}$ ( φ$_{j}$ ) + λF 2 ν φ$_{j}$ + λε 2 ν φ$_{j}$ ξ$_{j}$ ( t ) . (12) 7