First of all, we consider the situation that ω is so small that ( ψ dN dψ ) − 1 = ω − 1 has solu- tions. a ′ i has numerous kinks (discontinuities), from blunt ones to sharp ones, according to Eqs. (15), (16) and (19). Let us consider kinks which satisfy ( ψ dN dψ ) − 1 ≳ ω − 1 ( ⇔ ψ ≳ ψ$_{max}$ ). From now on, we call such kinks “big” kinks. The interval between two kinks with sharpness O ( ψ ) is roughly given by ( ψ dN dψ ) − $^{1}$. Thus the typical interval of big kinks is about ω − $^{1}$. First, we divide the integration range of Eq. (6) into short intervals of length ∼ ω − 1 around each big kink as where ¯ a ′ $^{i}$( u ) denotes the smooth function (except one big kink) which we can get after averaging contributions of small kinks to a ′ $^{i}$, and δa ′ i l is the contribution of small kinks. δa ′ i l discontinuously jumps at each small kink and the width of the jump is ∼ ψ 1 / $^{2}$. Its average vanishes ( 〈 δa ′ i l 〉 = 0) since the jump at each kink takes random values. Then we get after the integration of Eq. (B2). Then, Here the integral is performed over the l -th interval. We are interested in ensemble averages of products of two of I i $_{+}$, for example, 〈| I i + | $^{2}$〉 . It contains mean squares of I i + $_{,l}$’s and cross terms of different I i + $_{,l}$’s. First, we evaluate mean squares of I i + $_{,l}$’s. 〈| I i + ,l | $^{2}$〉 contains mean squares of the first and second terms in Eq. (B4), and the averages of the cross terms between them. The latter vanish since 〈 δa ′ i l 〉 = 0. In order to estimate the mean square of the first term in Eq. (B4), we approximate it as 22 a ′ $^{i}$( u ) = ¯ a ′ i l ( u ) + δa ′ i l ( u ) . (B2) a $^{i}$( u ) = ¯ a i $_{l}$( u ) + δa i $_{l}$( u ) , (B3) I i + $_{,l}$( k ) = ∫ l du ¯ a ′ i l exp( iω ( u − n · ¯ a$_{l}$ ( u ) − n · δ a$_{l}$ ( u )) / 2)+ ∫ l duδa ′ i l exp( iω ( u − n · ¯ a$_{l}$ ( u ) − n · δ a$_{l}$ ( u )) / 2) . (B4) ∫ l du ¯ a ′ i l exp( iω ( u − n · ¯ a$_{l}$ ( u ) − n · δ a$_{l}$ ( u )) / 2) ≃ ∑ a ¯ a ′ i l ( u$_{l}$$_{;}$$_{a}$ ) e iω ( u$_{l}$$_{;}$$_{a}$ − n · ¯ a ( u$_{l}$$_{;}$$_{a}$ )) / $^{2}$e − iω n · δ a ( u$_{l}$$_{;}$$_{a}$ ) / $^{2}$Δ u$_{l}$$_{;}$$_{a}$. (B5) I i $_{+}$( k ) = ∑ l I i + $_{,l}$( k ) , (B1) where the integer l labels each big kink and I i + $_{,l}$( k ) denotes the contribution to I i $_{+}$( k ) from the l -th interval. Each interval contains one big kink and numerous “small” kinks, which satisfy ( ψ dN dψ ) − 1 ≲ ω − 1 ( ⇔ ψ ≲ ψ$_{max}$ ). Let us assume that in the l -th interval a ′ i can be decomposed as