5 GeV/c Spill into B Line for Experiment 703

Finite-temperature and finite-time scaling of the directed polymer free energy with respect to its geometrical fluctuations

“…As suggested by the scaling relation (23), the slope of the linear initial growth of the free-energy correlator should fully account for the temperature dependence. As extensively discussed in [58], it can in fact be argued that the (D, T )-dependence of the scaling law (22) can be absorbed in a single prefactor D, which in the case of the discrete DP model reads…”

“…It is known from the large-time limit at zero ξ that D = cD T , see (17) (and also (C25) in the diffusive regime). In the discrete DP case (c = T ), D is thus expected to be temperature independent in the high temperature regime, D = D, and to behave as T 2/3 in the low temperature regime [58]. In order to test this, we rescale the data in Fig.…”

“…With this choice of normalization, it is known from the Feynman-Kac formula [54][55][56][57] (see also Ref. [58] for a discussion) that the weight Z V (t, y) evolves according to the 'stochastic heat equation' [59][60][61]…”

“…Ref. [58] for a derivation at non-zero ξ): where c y 2 2t + T 2 log 2πT t c is the elastic contribution, which fully captures the initial condition, whileF V (t, y) is invariant by translation along y in distribution (in mathematical terms [49]F V (t, y) is 'stationary' along y). It represents the contribution of the disordered potential to the free-energy (F V (t, y)| V ≡0 = 0).…”

“…(see also Refs. [53,58] when the disorder correlation length ξ is nonzero). We propose that t ∼ B(t) in (19), in other words…”