content_image/354217.jpg content_image/354216.jpg content_image/354219.jpg
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, 

<img src='content_image/354217.jpg'>

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 

<img src='content_image/354216.jpg'>

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, 

<img src='content_image/354219.jpg'>

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 

