Large-time asymptotics of the gyration radius for long-range statistical-mechanical models
aa r X i v : . [ m a t h . P R ] D ec Large-time asymptotics of the gyration radius forlong-range statistical-mechanical models ∗ Akira Sakai † October 13, 2018
Abstract
The aim of this short article is to convey the basic idea of the original paper [3],without going into too much detail, about how to derive sharp asymptotics of thegyration radius for random walk, self-avoiding walk and oriented percolation abovethe model-dependent upper critical dimension.
Let D be the Z d -symmetric 1-step distribution for random walk (RW) and define the RW2-point function as ϕ RW t ( x ) = X ω : o → x | ω | = t t Y s =1 D ( ω s − ω s − ) ( x ∈ Z d , t ∈ Z + ) . (1.1)We also consider self-avoiding walk (SAW) and oriented percolation (OP) that are bothgenerated by D . The SAW 2-point function is defined as ϕ SAW t ( x ) = X ω : o → x | ω | = t t Y s =1 D ( ω s − ω s − ) Y ≤ i RIMS Kokyuroku Bessatsu . † Creative Research Institution SOUSEI, Hokkaido University, Japan. [email protected] p equals the expected number of occupied bonds per vertex, andit is known that there is a phase transition at p = p c . We say that ( x, s ) is connected to( y, t ) if either ( x, s ) = ( y, t ) or there is a time-increasing sequence of occupied bonds from( x, s ) to ( y, t ). The OP 2-point function ϕ OP t ( x ) is then defined as the probability thatthe origin ( o, 0) is connected to ( x, t ).The models are said to be finite-range if D is supported on a finite set of Z d . The mainproperty of a finite-range D is the existence of the variance σ ≡ P x ∈ Z d | x | D ( x ), andbecause of this, investigation of finite-range models is relatively easier. The situation isbasically the same for D that decays faster than any polynomials, such as an exponentiallydecaying D . However, if D ( x ) ≈ | x | − d − α for large | x | , then the existence of the variancedepends on α > α . For example, take thegyration radius of order r ∈ (0 , α ), which is defined as ξ ( r ) t = (cid:18) P x ∈ Z d | x | r ϕ t ( x ) P x ∈ Z d ϕ t ( x ) (cid:19) /r . (1.3)The gyration radius represents a typical end-to-end distance of a linear structure of length t or a typical spatial size of a cluster at time t . It may be natural to guess, at least forrandom walk, that ξ ( r ) t = O ( √ t ) if α > ξ ( r ) t = O ( t /α ) if α < 2, for every real r ∈ (0 , α ). As we state shortly, we have proved affirmative results [3] for random walkin any dimension and for self-avoiding walk and critical/subcritical oriented percolationabove the common upper-critical dimension d c ≡ α ∧ D . Given an L ∈ [1 , ∞ ), wesuppose that D ( x ) ∝ | x/L | − d − α for large | x | such that its Fourier transform ˆ D ( k ) ≡ P x ∈ Z d e ik · x D ( x ) exhibits the k → − ˆ D ( k ) = v α | k | α ∧ × ( O (( L | k | ) ǫ ) ( α = 2) , log L | k | + O (1) ( α = 2) , (1.4)for some v α = O ( L α ∧ ) and ǫ > 0. If α > D is finite-range), then v α = d σ . Anexample that satisfies the above properties is the long-range Kac potential D ( x ) = h ( y/L ) P y ∈ Z d h ( y/L ) ( x ∈ Z d ) , (1.5)defined by the rotation-invariant function h ( x ) = 1 + O (cid:0) ( | x | ∨ − ρ (cid:1) ( | x | ∨ d + α ( x ∈ R d ) , (1.6)for some ρ > ǫ (cf., [3]). Under this assumption, we have proved the following sharpasymptotics of a variant of the gyration radius:2 heorem 1.1 ([3]) . For random walk in any dimension with any L , and for self-avoidingwalk and critical/subcritical oriented percolation for d > d c with L ≫ , there is a model-dependent constant C α = 1 + O ( L − d ) ( C α ≡ for random walk) such that, for every r ∈ (0 , α ) , P x ∈ Z d | x | r ϕ t ( x ) P x ∈ Z d ϕ t ( x ) ∼ t ↑∞ rπα ∨ ( α ∧ 2) sin rπα Γ( r + 1)Γ( rα ∧ + 1) × ( ( C α v α t ) rα ∧ ( α = 2) , ( C v t log √ t ) r/ ( α = 2) , (1.7) where x is the first coordinate of x ∈ Z d . We should emphasize that, except for the actual value of C α , the expression (1.7) isuniversal. The result also holds for finite-range models, for which α is considered to beinfinity. As far as we notice, even for random walk, the sharp asymptotic expression (1.7)for all real r ∈ (0 , α ) is new.Using | x | r ≤ | x | r ≤ d r/ P dj =1 | x j | r and the Z d -symmetry of the models, we canconclude the following: Corollary 1.2 ([3]) . Under the same condition as in Theorem 1.1, ξ ( r ) t = ( O ( t α ∧ ) ( α = 2) ,O ( √ t log t ) ( α = 2) . (1.8) for every r ∈ (0 , α ) . In his recent work [4], Heydenreich proved (1.8) for self-avoiding walk, but only forsmall r < α ∧ 2, with no attempt to identify the proportional constant. Our results aresomewhat stronger, because we have derived the exact expression for the proportionalconstant in (1.7) (also clarifying its model-dependence) and proved (1.8) for all r < α . In this section, we restrict our attention to random walk, which is obviously simpler thanthe other two models, and explain the framework of the proof of Theorem 1.1.First we consider the generating function (= the Fourier-Laplace transform) of the2-point function. Recall that ϕ RW t ( x ) satisfies the convolution equation ϕ RW t ( x ) = δ t, δ x,o + ( D ∗ ϕ RW t − )( x ) ≡ δ t, δ x,o + X y ∈ Z d D ( y ) ϕ RW t − ( x − y ) , (2.1)where we regard ( D ∗ ϕ RW t − )( x ) for t ≤ k ∈ [ − π, π ] d and m ∈ [0 , m RW c ),ˆ ϕ RW m ( k ) ≡ X t ∈ Z + m t X x ∈ Z d e ik · x ϕ RW t ( x ) = 1 + m ˆ D ( k ) ˆ ϕ RW m ( k ) , (2.2)3here m RW c ≡ (cid:8) P x ∈ Z d ϕ RW t ( x ) (cid:9) t ∈ Z + . Tosee this in a different way, take k = 0 in (2.2) so thatˆ ϕ RW m (0) = 1 + m ˆ ϕ RW m (0) = 11 − m . (2.3)The expansion of the right-hand side is P t ∈ Z + m t and the coefficient of m t is exactly 1( ≡ P x ∈ Z d ϕ RW t ( x )) for every t ∈ Z + .Next we differentiate ˆ ϕ RW m ( k ) with respect to k (= the first coordinate of k ) to yieldthe generating function of the sequence (cid:8) P x ∈ Z d | x | r ϕ RW t ( x ) (cid:9) t ∈ Z + . For example, if r = 2 j with j ∈ N (hence α > ∇ j ˆ ϕ RW m (0) ≡ ∂ j ∂k j ˆ ϕ RW m ( k ) (cid:12)(cid:12)(cid:12)(cid:12) k =0 = ( − j X t ∈ Z + m t X x ∈ Z d x j ϕ RW t ( x ) . (2.4)On the other hand, by differentiating (2.2) and using the Z d -symmetry of the model, ∇ j ˆ ϕ RW m (0) = m ∇ j ˆ ϕ RW m (0) + m j X l =1 (cid:18) j l (cid:19) ∇ l ˆ D (0) ∇ j − l )1 ˆ ϕ RW m (0)= m − m j X l =1 (cid:18) j l (cid:19) ∇ l ˆ D (0) ∇ j − l )1 ˆ ϕ RW m (0) . (2.5)Solving this recursion by induction under the initial condition (2.3), we obtain (see [3] formore details) ∇ j ˆ ϕ RW m (0) = (cid:18) j (cid:19) m ∇ ˆ D (0)1 − m ∇ j − ˆ ϕ RW m (0) + O (cid:0) (1 − m ) − j (cid:1) = (cid:18) j (cid:19)(cid:18) j − (cid:19)(cid:18) m ∇ ˆ D (0)1 − m (cid:19) ∇ j − ˆ ϕ RW m (0) + O (cid:0) (1 − m ) − j (cid:1) ...= j Y l =1 (cid:18) l (cid:19)(cid:18) m ∇ ˆ D (0)1 − m (cid:19) j ˆ ϕ RW m (0) + O (cid:0) (1 − m ) − j (cid:1) = (2 j )!2 j (cid:0) m ∇ ˆ D (0) (cid:1) j (1 − m ) j +1 + O (cid:0) (1 − m ) − j (cid:1) . (2.6)Comparing this with (2.4) and using v α ≡ d σ = − ∇ ˆ D (0) for α > 2, we arrive at X t ∈ Z + m t X x ∈ Z d x j ϕ RW t ( x ) = (2 j )! ( mv α ) j (1 − m ) j +1 + O (cid:0) (1 − m ) − j (cid:1) . (2.7)4owever, by the general binomial expansion, m j (1 − m ) j +1 = m j ∞ X l =0 (cid:18) − j − l (cid:19) ( − m ) l = m j ∞ X l =0 (cid:18) j + lj (cid:19) m l = ∞ X t = j (cid:18) tj (cid:19) m t . (2.8)Therefore, X x ∈ Z d x j ϕ RW t ( x ) ∼ t ↑∞ (2 j )! (cid:18) tj (cid:19) v jα ∼ Γ(2 j + 1)Γ( j + 1) ( v α t ) j . (2.9)This completes the proof of (1.7) for r = 2 j < α .In order to consider the other values of r < α , we use the following integral represen-tation for | x | q with q ∈ (0 , 2) (cf., [3]): | x | q = 1 K q Z ∞ − cos( ux ) u q d u, (2.10)where K q = Z ∞ − cos uu q d u = π qπ q + 1) . (2.11)Let r = 2 j + q with j ∈ Z + and q ∈ (0 , (cid:8) P x ∈ Z d | x | j + q ϕ RW t ( x ) (cid:9) t ∈ Z + can be written as X t ∈ Z + m t X x ∈ Z d | x | j + q ϕ RW t ( x ) = 1 K q Z ∞ d uu q X t ∈ Z + m t X x ∈ Z d (cid:0) − cos( ux ) (cid:1) x j ϕ RW t ( x )= ( − j K q Z ∞ d uu q (cid:16) ∇ j ˆ ϕ RW m (0) − ∇ j ˆ ϕ RW m ( ~u ) (cid:17) , (2.12)where ~u = ( u, , . . . , ∈ R d . Therefore, similarly to the above case of r = 2 j , it sufficesto investigate the “derivative”¯∆ ~u ∇ j ˆ ϕ RW m (0) ≡ ∇ j ˆ ϕ RW m (0) − ∇ j ˆ ϕ RW m ( ~u ) . (2.13)However, by “differentiating” both sides of (2.2) and using the Z d -symmetry, we obtain¯∆ ~u ∇ j ˆ ϕ RW m (0) = m ¯∆ ~u ∇ j ˆ ϕ RW m (0) + m j X l =1 (cid:18) j l (cid:19) ∇ l ˆ D (0) ¯∆ ~u ∇ j − l )1 ˆ ϕ RW m (0)+ m j X n =0 (cid:18) jn (cid:19) ∇ j − n ˆ ϕ RW m ( ~u ) ¯∆ ~u ∇ n ˆ D (0)= m − m (cid:18) j X l =1 (cid:18) j l (cid:19) ∇ l ˆ D (0) ¯∆ ~u ∇ j − l )1 ˆ ϕ RW m (0)+ j X n =0 (cid:18) jn (cid:19) ∇ j − n ˆ ϕ RW m ( ~u ) ¯∆ ~u ∇ n ˆ D (0) (cid:19) , (2.14)5here we regard the sum over l ∈ { , . . . , j } in the last expression as zero when j = 0.Substituting this back to (2.12), performing the integration with respect to u ∈ (0 , ∞ )and then reorganizing the resulting terms (see [3] for more details), we will end up with X t ∈ Z + m t X x ∈ Z d | x | r ϕ RW t ( x ) = 2 sin rπα ∨ ( α ∧ 2) sin rπα Γ( r + 1) ( mv α ) rα ∧ (1 − m ) rα ∧ × O ((1 − m ) ǫ ) ( α = 2) , (cid:0) log √ − m (cid:1) r/ + O (1) ( α = 2) , (2.15)for some ǫ > 0. The proof of (1.7) is completed by expanding the right-hand side of theabove expression in powers of m and comparing the coefficient of m t in both sides, forlarge t . The key to the proof for self-avoiding walk and oriented percolation is the following laceexpansion (see, e.g., [1, 5]): ϕ t ( x ) = I t ( x ) + t X s =1 ( J s ∗ ϕ t − s )( x ) , (3.1)where I t ( x ) = ( δ x,o δ t, (SAW) ,π OP t ( x ) (OP) , J t ( x ) = ( D ( x ) δ t, + π SAW t ( x ) (SAW) ,p ( D ∗ π OP t − )( x ) (OP) . (3.2)Recall (2.1) for random walk, so that I RW t ( x ) = δ x,o δ t, and J RW t ( x ) = D ( x ) δ t, . Themodel-dependent π t ( x ) in (3.2), which accounts for difference from random walk, is analternating sum of the lace-expansion coefficients and obey the following diagrammaticbounds (cf., [1, 5]): | π SAW t ( x ) | ≤ x = o + xo + o x + · · · , (3.3) | π OP t ( x ) | ≤ ( x,t )( o, + ( x,t )( o, + ( x,t )( o, + ( x,t )( o, + · · · , (3.4)where each line corresponds to a 2-point function. For self-avoiding walk, the first diagramrepresents self-avoiding loop of length t ≥ 2, i.e., ( D ∗ ϕ SAW t − )( x ), and the second diagram6epresents the product of three 2-point functions, ϕ SAW s ( x ) ϕ SAW s ( x ) ϕ SAW s ( x ), summed overall possible combinations of s , s , s ≥ s + s + s = t , and so on. Fororiented percolation, the first diagram represents ϕ OP t ( x ) , where the upward directionis the time-increasing direction, and the second diagram represents the product of five2-point functions concatenated in a depicted way, where unlabeled vertices are summedover Z d × Z + , and so on. For more details, we refer to [1, 5].Because of the similarity between (2.1) and (3.1), it is natural to expect that thestrategy in § k ∈ [ − π, π ] d and m ∈ [0 , m c ), ˆ ϕ m ( k ) = ˆ I m ( k ) + ˆ J m ( k ) ˆ ϕ m ( k ) , (3.5)where m c ≥ (cid:8) P x ∈ Z d ϕ t ( x ) (cid:9) t ∈ Z + forself-avoiding walk and critcal/subcritical oriented percolation ( m OP c is a non-increasingfunction of p ≤ p c and m OP c = 1 at p = p c [1]). Due to the diagrammatic bounds (3.3)–(3.4), it has been proved [1, 2, 4] that, for d > d c and L ≫ 1, there are ǫ, δ > X t ∈ Z d t ǫ m t X x ∈ Z d | π t ( x ) | , X t ∈ Z d m t X x ∈ Z d | x | α ∧ δ | π t ( x ) | , (3.6)both converge, even at m = m c . This implies that ˆ J m c (0) = 1 and, as m ↑ m c ,ˆ ϕ m (0) = ˆ I m (0)1 − ˆ J m (0) = ˆ I m (0)ˆ J m c (0) − ˆ J m (0) ∼ ˆ I m c (0) m c ∂ m ˆ J m c (0) (cid:0) − mm c (cid:1) = ˆ I m c (0) m c ∂ m ˆ J m c (0) X t ∈ Z + (cid:16) mm c (cid:17) t . (3.7)On the other hand, for r = 2 j < α with j ∈ N , ∇ j ˆ ϕ m (0) = ∇ j ˆ I m (0) + j X l =0 (cid:18) j l (cid:19) ∇ l ˆ J m (0) ∇ j − l )1 ˆ ϕ m (0)= 11 − ˆ J m (0) (cid:18) ∇ j ˆ I m (0) + j X l =1 (cid:18) j l (cid:19) ∇ l ˆ J m (0) ∇ j − l )1 ˆ ϕ m (0) (cid:19) . (3.8)Suppose that the leading contribution is due to the l = 1 term (this is far from trivial7nd needs to be proved, as in [3]). Then, by induction and using (3.7), ∇ j ˆ ϕ m (0) ∼ (cid:18) j (cid:19) ∇ ˆ J m (0)1 − ˆ J m (0) ∇ j − ˆ ϕ m (0)... ∼ (2 j )!2 j (cid:18) ∇ ˆ J m (0)1 − ˆ J m (0) (cid:19) j ˆ ϕ m (0) ∼ (2 j )!2 j (cid:18) ∇ ˆ J m c (0) m c ∂ m ˆ J m c (0) (cid:0) − mm c (cid:1) (cid:19) j ˆ I m c (0) m c ∂ m ˆ J m c (0) (cid:0) − mm c (cid:1) . (3.9)However, similarly to (2.8), (cid:16) − mm c (cid:17) − j − = X t ∈ Z + (cid:18) t + jj (cid:19)(cid:16) mm c (cid:17) t , (3.10)hence ∇ j ˆ ϕ m (0) ∼ (2 j )! (cid:18) ∇ ˆ J m c (0)2 m c ∂ m ˆ J m c (0) (cid:19) j ˆ I m c (0) m c ∂ m ˆ J m c (0) X t ∈ Z + (cid:18) t + jj (cid:19)(cid:16) mm c (cid:17) t . (3.11)Therefore, by (3.7) and (3.11), P x ∈ Z d x j ϕ t ( x ) P x ∈ Z d ϕ t ( x ) ∼ (2 j )! j ! (cid:18) −∇ ˆ J m c (0)2 m c ∂ m ˆ J m c (0) t (cid:19) j = Γ(2 j + 1)Γ( j + 1) (cid:18) m c ∂ m ˆ J m c (0) ∇ ˆ J m c (0) ∇ ˆ D (0) | {z } C α −∇ ˆ D (0)2 | {z } v α t (cid:19) j . (3.12)This completes a sketch proof for r = 2 j .The case for the other values of r < α is more involved, but can be proved by followingthe same strategy as in § C α in (3.12) is ill-defined for α ≤ ∇ ˆ D (0), it is replaced by C α = 1 m c ∂ m ˆ J m c (0) lim k → ¯∆ k ˆ J m c (0)¯∆ k ˆ D (0) ≡ m c ∂ m ˆ J m c (0) lim k → ˆ J m c (0) − ˆ J m c ( k )ˆ D (0) − ˆ D ( k ) . (3.13)We refrain from showing further details and refer the readers to the original paper [3]. Acknowledgements This work was supported by the start-up fund of the Leader Development System in theBasic Interdisciplinary Research Areas at Hokkaido University. I am grateful to Lung-Chi8hen for the fruitful collaboration on the long-range models [1, 2, 3] and Keiichi R. Itofor the invitation to the RIMS workshop “Applications of RG Methods in MathematicalSciences” at Kyoto University from September 9 through 11 th , 2009. References [1] L.-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-rangeoriented percolation. I. Probab. Theory Relat. Fields (2008): 151–188.[2] L.-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-rangeoriented percolation. II: Spatial correlation. Probab. Theory Relat. Fields (2009):435–458.[3] L.-C. Chen and A. Sakai. Asymptotic behavior of the gyration radius for long-rangeself-avoiding walk and long-range oriented percolation. In preparation.[4] M. Heydenreich. Long-range self-avoiding walk converges to α -stable processes.Preprint, arXiv:0809.4333v1 (2008).[5] G. Slade. The lace expansion and its applications. Lecture Notes in Math.1879