3 γ ( t ) near the end of the cooling procedure is exponential rather than algebraic we conclude that This suggests to analyze a family of cooling laws with an asymptotic behavior of γ ( t ) of the form Thus to determine G$_{1}$ we need to solve an integral equation with arbitrary positive parameters B, b, β . In this sit- uation, the asymptotic behavior of the final density of domain walls is Higher correlation functions G$_{k}$ with k ≥ 2 are then ex- pressed via G$_{1}$ and (18). The above results are exact. Now we turn to the most interesting case of slow cooling, τ ≫ 1, and compute the final density of domain walls. This is a remarkably universal asymptotic: The leading τ − 1 / 2 algebraic factor does not depend on the parameters B, b, β and the logarithmic prefactor depends only on one parameter β . We now turn to derivations of the announced results. Using (4), we re-write the integral equation (19) in terms of the density of domain walls: In this section we consider the special cooling proce- dure (6). First, we employ an exact analysis and then turn to the asymptotic behavior of the final density of domain walls. We are mostly interested in the final density ρ ( τ ). Hence we specialize (20) to t = τ . It is also convenient to change the integration variable, t ′ → η = ( τ 2 − t ′ $^{2}$) /τ . We arrive at the integral equation To handle an infinite set of ordinary differential equa- tions (8) we recast it into a single differential equation using a generating function technique. Multiplying (8) by z k and summing over all k ≥ 1 we find that the gen- erating function satisfies Solving (17) subject to G ( z, t = 0) = 0, we get where η = ( t 2 − t ′ $^{2}$) /τ . Recalling that the exponential factor exp [ η z + z 1 2 ] is the generating function for the modified Bessel functions as it follows by expanding the term inside the square brackets in Eq. (22). Thus the integral on the right-hand side of (21) becomes We can further use the asymptotic formula for the Bessel function, The chief contribution to the integral on the right-hand side of (21) is gathered when η ∼ √$_{τ}$ [see Eq. (26) below]. Therefore we can replace ρ ( τ √ 1 − η/τ ) by ρ ( τ ) and the exponential factor E ( η, τ ) by where we used the shorthand notation G ( z, t ) = ∑ k ≥ 1 G$_{k}$ ( t ) z k (16) E ( η, τ ) ≡ exp [ 2 τ ( √ 1 − η/τ − 1 )] (22) 1 − 1 2 ∫ τ 0 dη E ( η, τ ) [ I$_{0}$ ( η ) − I$_{1}$ ( η )] = 2 ρ ( τ ) + ∫ τ 0 dη E ( η, τ ) I$_{1}$ ( η ) ρ ( τ √ 1 − η/τ ) (21) E ( η, τ ) = e − η − η $^{2}$/ 4 τ (23) ρ ( τ ) ∫ τ 0 dη e − η − η $^{2}$/ 4 τ I$_{1}$ ( η ) (24) I$_{1}$ ( η ) ≃ e η √$_{2}$$_{πη}$ when η ≫ 1 , (25) exp [ η z + z − 1 2 ] = ∞ ∑ n = −∞ z n I$_{n}$ ( η ) G = ∫ t 0 dt ′ t ′ τ [ z − G$_{1}$ ( t $^{′}$)] exp [ 2( t ′ − t ) + η z + z − 1 2 ] ∂G ∂t = [ − 2 + t τ ( z + z − $^{1}$) ] G + t τ [ z − G$_{1}$ ( t )] (17) 1 − ∫ t 0 dt ′ t ′ τ e 2( t $^{′}$− t $^{)}$[ I$_{0}$ ( η ) − I$_{1}$ ( η )] = 2 ρ ( t ) + 2 ∫ t 0 dt ′ t ′ τ e 2( t $^{′}$− t ) I$_{1}$ ( η ) ρ ( t $^{′}$) (20) ρ ( τ ) ≃ 1 4 √ π τ ( ln τ b ) 1 2 β (15) 1 − γ ( t ) ≃ B exp { − b (1 − t/τ ) β } when t → τ (14) 1 − γ ( t ) ≃ 2 exp { − 4 /T$_{0}$ 1 − t/τ } when t → τ G$_{k}$ ( t ) = ∫ t 0 dt ′ t ′ τ e 2( t $^{′}$− t $^{)}$[ I$_{k}$$_{−}$$_{1}$ ( η ) − G$_{1}$ ( t $^{′}$) I$_{k}$ ( η )] (18) G$_{1}$ ( t ) = ∫ t 0 dt ′ t ′ τ e 2( t $^{′}$− t $^{)}$[ I$_{0}$ ( η ) − G$_{1}$ ( t $^{′}$) I$_{1}$ ( η )] (19) B. Final Density of Domain Walls III. SPECIAL COOLING SCHEME A. Exact Analysis