Here we write the position of the a -th small kink in the l -th interval as u$_{l}$$_{;}$$_{a}$ , and Δ u$_{l}$$_{;}$$_{a}$ = u$_{l}$$_{;}$$_{a}$$_{+1}$ − u$_{l}$$_{;}$$_{a}$ . We assume that each interval between two small kinks is so short that the integrand can be regarded as constant. We want to evaluate the mean square of this quantity, Here we separate the average related to ¯ a ′ i l , ¯ a i l and that related to δa ′ i l , δa i $_{l}$, assuming that there is no correlation between small kinks and big kinks. In order to evaluate the second parenthesis in Eq. (B6), we decompose δa i l as To proceed from RHS of the first line to the second line, we regard F ( k ) i l ( u ) as constant in the interval between two small kinks. Here u k l,s denotes the position of s -th discontinuity of F ( k ) i l ( u ). F ( k ) i l ( u k $_{l,s}$) can be thought of as a probability variable whose average is 0 and whose variance is ∼ ψ$_{k}$ . We can set 〈 F ( k ) i l ( u k $_{l,s}$) F ( k $^{′}$) i l ( u k ′ l,s ′ ) 〉 = 0 unless k = k $^{′}$, s = s $^{′}$, assuming 23 δa ′ i l = ∑ k F ( k ) i l ( u ) , (B7) ∑ a,b 〈 ¯ a ′ i l ( u$_{l}$$_{;}$$_{a}$ )¯ a ′ i l ( u$_{l}$$_{;}$$_{b}$ ) e iω ( u$_{l}$$_{;}$$_{a}$ − n · ¯ a ( u$_{l}$$_{;}$$_{a}$ )) / $^{2}$e − iω ( u$_{l}$$_{;}$$_{b}$ − n · ¯ a ( u$_{l}$$_{;}$$_{b}$ )) / $^{2}$〉〈 e − iω n · δ a ( u$_{l}$$_{;}$$_{a}$ ) / $^{2}$e iω n · δ a ( u$_{l}$$_{;}$$_{b}$ ) / $^{2}$〉 Δ u$_{l}$$_{;}$$_{a}$ Δ u$_{l}$$_{;}$$_{b}$ (B6) where F ( k ) i l ( u ) is the contribution of kinks of sharpness ψ ∼ ψ$_{k}$ , so it has discontinuities at intervals ∼ ( ψ$_{k}$ dN $_{dψ}$( ψ$_{k}$ ) ) − 1 and between two of them its absolute value ∼ ψ 1 / 2 k . Then, [ 〈| δ a$_{l}$ ( u$_{l}$$_{;}$$_{a}$ ) − δ a$_{l}$ ( u$_{l}$$_{;}$$_{b}$ ) | $^{2}$〉 ] 1 / 2 ∼ [ 〈| δa i $_{l}$( u$_{l}$$_{;}$$_{a}$ ) − δa i $_{l}$( u$_{l}$$_{;}$$_{b}$ ) | $^{2}$〉 ] 1 / 2 ∼   〈( ∑ k ∑ s F ( k ) i l ( u k $_{l,s}$)( u k l,s +1 − u k $_{l,s}$) ) $^{2}$〉  $^{1}$$^{/}$$^{2}$ ∼ [ ∑ k ∑ s 〈 ( F ( k ) i l ( u k $_{l,s}$)) $^{2}$〉 ( u k l,s +1 − u k $_{l,s}$) 2 ] 1 / 2 ∼ [ ∑ k ψ$_{k}$ × ( ψ$_{k}$ dN dψ ( ψ$_{k}$ ) ) − 2 × ( | u$_{l}$$_{;}$$_{a}$ − u$_{l}$$_{;}$$_{b}$ | / ( ψ$_{k}$ dN dψ ( ψ$_{k}$ ) ) − 1 )] 1 / 2 ∼ [ ∑ k ψ$_{k}$ ( ψ$_{k}$ dN dψ ( ψ$_{k}$ ) ) − 1 ] 1 / 2 | u$_{l}$$_{;}$$_{a}$ − u$_{l}$$_{;}$$_{b}$ | 1 / 2 ∼ [ ∫ ψ$_{max}$ dψψ − 1 ( dN dψ ( ψ ) ) − 1 ] 1 / 2 | u$_{l}$$_{;}$$_{a}$ − u$_{l}$$_{;}$$_{b}$ | 1 / 2 ∼ ( dN dψ ( ψ$_{max}$ ) ) − 1 / 2 | u$_{l}$$_{;}$$_{a}$ − u$_{l}$$_{;}$$_{b}$ | 1 / $^{2}$. (B8)