Critical two-point function for long-range models with power-law couplings: The marginal case for d d c

Transcription

1 Critical two-point function for long-range models with power-law couplings: The marginal case for d d c ung-chi Chen, Akira Sakai arxiv: v3 [math-ph] 17 Jan 19 January 18, 19 Abstract Consider the long-range models on Z d of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as x d α for some α >. In the previous work [15], we have shown in a unified fashion for all α that, assuming a bound on the derivative of the n-step distribution the compound-zeta distribution satisfies this assumed bound, the critical two-point function G pc x decays as x α d above the uppercritical dimension d c α m, where m = for self-avoiding walk and the Ising model and m = 3 for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the n-step distribution, that G pc x for the marginal case α = decays as x d / log x whenever d d c with a large spread-out parameter. This solves the conjecture in [15], extended all the way down to d = d c, and confirms a part of predictions in physics [1]. The proof is based on the lace expansion and new convolution bounds on power functions with log corrections. Contents 1 Introduction and the main results 1.1 Introduction The models and the main results Random walk Self-avoiding walk Percolation The Ising model The main results Department of Mathematical Sciences, National Chengchi University, Taiwan Mathematics Division, National Center for Theoretical Sciences, Taiwan Faculty of Science, Hokkaido University, Japan. 1

2 Analysis for the underlying random walk 1.1 Proof of Theorem Proof of the bound 1.1 on D n for α = Analysis for the two-point function ist of known facts Proof of the infrared bound Convolution bounds on power functions with log corrections Bounds on the lace-expansion coefficients Proof of the asymptotic behavior Introduction and the main results 1.1 Introduction The lace expansion has been successful in rigorously proving mean-field critical behavior for various models, such as self-avoiding walk [11], percolation [19], lattice trees and lattice animals [], oriented percolation [6], the contact process [7], the classical Ising and ϕ 4 models [8, 9]. It provides a way to derive a formal recursion equation for the two-point function G p x, which is similar to the recursion equation for the random-walk Green function S p x generated by the non-degenerate i.e., Do < 1 1-step distribution Dx and the fugacity p [, 1]: S p x = δ o,x + pd S p x, 1.1 where, and in the rest of the paper, f gx y fy gx y is the convolution of two functions f, g on Z d. The formal recursion equation for G p x is of the form G p x = Π p x + Π p pd G p x, 1. where Π p x is a series of the model-dependent lace-expansion coefficients. It is natural to expect that, once regularity of Π p e.g., absolute summability is assured for all p up to the critical point p c, the asymptotic behavior of G pc x should be the same modulo constant multiplication as that for the random-walk Green function S 1 x. If so, then sufficient conditions for the mean-field behavior, called the bubble condition for selfavoiding walk and the Ising model [1, 4] and the triangle condition for percolation [6], hold for all dimensions above the model-dependent upper-critical dimension d c, which is m for short-range models, where m = for self-avoiding walk and the Ising model and m = 3 for percolation. In recent years, long-range models defined by power-law couplings, Dx x d α for some α >, have attracted more attention, due to unconventional critical behavior and crossover phenomena e.g., [7, 1, 15, 3]. Under some mild assumptions, we have shown [15, Proposition.1] that, for α and d > α, the random-walk Green function S 1 x

3 is asymptotically γα v α x α d, where γ α = Γ d α α π d/ Γ α, v 1 α = lim ˆDk k k α 1 eik x lim Dx. 1.3 k k α x Z d For short-range models with variance σ = x x Dx <, the asymptotic behavior of S 1 x is well-known to be d d Γ π d/ σ x d, which is consistent with 1.3 for large α >. The crossover occurs at α =, where the variance σ diverges logarithmically and S 1 x was believed to have a log correction to the above standard Newtonian behavior. An example of Dx x d α is the compound-zeta distribution see 1.15 for the precise definition. It has been shown [15] that this long-range distribution for α also satisfies a certain bound on the derivative D n x 1 D n x + y + D n x y of the n-step distribution. Thanks to this extra bound, we have shown [15, Theorem 1.] in a unified fashion for all α that, whenever d > d c α m with a large spread-out parameter, there is a model-dependent constant A close to 1 in fact, A = 1 for α < such that G pc x A p c S 1 x. One of the key elements to showing this result is slight improvement of the convolution bounds on power functions [18, Proposition 1.7] that are used to prove regularity of Π p in 1.. However, since those convolution bounds are not good enough to properly control power functions with log corrections, we were unable to achieve an asymptotic result for α =, until the current work. In this paper, we tackle the marginal case α =. The headlines are the following: S 1 x γ v x d / log x whenever d >, where γ is in 1.3, but v is redefined as 1 v = lim ˆDk k k log1/ k. 1.4 G pc x 1 p c S 1 x whenever d d c with a large spread-out parameter. This also implies that other critical exponents take on their mean-field values for d d c including equality. The latter solves the conjecture [15, 1.9], extended all the way down to d = d c. It also confirms a part of predictions in physics [1, 3]: the critical two-point function for percolation was proposed to decay as x α η d whenever α η, where η = ηd is the anomalous dimension for short-range percolation and is believed to be nonzero for d < 6, and as x η d / log x whenever α = η. We should emphasize that the proof of the asymptotic result in this paper is rather different from the one in [15] for α. In fact, we do not require the n-step distribution D n to satisfy the aforementioned derivative bound. Because of this, we can cover a wider class of models to which the same result applies, and can simplify the proof to some extent. Although the same proof works for α < see Remark 3.7 below, we will focus on the marginal case α =. Before closing this subsection, we remark on recent progress in the renormalization group analysis for the On model, which is equivalent to self-avoiding walk when n = 3

4 and to the n-component ϕ 4 model when n 1. Suppose that the above physics prediction is true for the On model as well, and that η > for d < 4. Then, we can take a small ε > to satisfy α = d+ε d, η, hence d = α ε < d c, and yet G pc x is proven to decay as x α d [3]. This sticking at the mean-field behavior, even below the upper-critical dimension, has been proven by using a rigorous version of the ε-expansion. In the next subsection, we give more precise definitions of the concerned models. 1. The models and the main results 1..1 Random walk et x r = π x r [x Rd, 1 r < ], 1.5 where is the Euclidean norm. We require the 1-step distribution Dx to be bounded as Dx 1 x d d α 1 def c >, x Z d, [1, : c Dx 1 x d d α 1 1 c, 1.6 where is the spread-out parameter. et ˆD and D n be the Fourier transform and the n-fold convolution of D, respectively: ˆDk = x Z d e ik x Dx [k [ π, π d ], 1.7 D n x = { δ o,x [n = ], y Z d D n 1 y Dx y [n 1]. We also require D to satisfy the following properties. Assumption 1.1 Properties of ˆD. There is a =, 1 such that { 1 ˆDk < [ k [ π, π] d ], > [ k > 1/], while, for k 1/, 1 ˆDk k α { 1 [α ], log π [α = ]. k Moreover, there is an ɛ > such that, as k, { 1 ˆDk 1 + O ɛ = v α k α k ɛ [α ], log 1 + O1 [α = ], k where the constant in the O1 term is independent of

5 Assumption 1. Bounds on D n. For n N and x Z d, { D n O d n d/α [α ], n log πn d/ [α = ], { D n x n Oα 1 [α ], x d+α log x 1 [α = ] It has been shown [1, 14, 15] that the following D is one of the examples that satisfy all the properties in the above assumptions: x d α [x o], Dx = y Z d \{o} y d α 1.14 [x = o]. Another such example is the following compound-zeta distribution [15]: Dx = t N U t x T α t [x Z d ], 1.15 where, with a probability distribution h on [ 1, 1] d R d and the Riemann-zeta function ζs = t N t s, U x = T α t = hx/ y Z d \{o} hy/ [x Z d ], 1.16 t 1 α/ ζ1 + α/ [t N] We assume that the distribution h is bounded, non-degenerate, Z d -symmetric and piecewise continuous, such as hx = d 1 { x 1}. Since the proof of 1.1 for α = is only briefly explained in [15, 1.19], we will provide a full proof in Section. et S p be the random-walk Green function generated by the 1-step distribution D: S p x = ω:o x p ω ω Dω j ω j 1 [x Z d ], 1.18 j=1 where o Z d is the origin, p is the fugacity and ω is the length of a path ω = ω, ω 1,..., ω ω. By convention, the contribution from the zero-step walk is the Kronecker delta δ o,x. It is convergent as long as p < 1 or p = 1 with d > α. One of the main results of this paper is completion of the asymptotic picture of S 1 for all α >, as follows. Theorem 1.3. et d > α and suppose D satisfies Assumptions Then, for any p [, 1], S p x δ o,x O α 1 [α ], x d α log x [α = ]. 1 5

6 Moreover, there are ɛ, η > such that, for 1+η < x, S 1 x = γ 1 + Oɛ α/v α x x ɛ d α O1 log x log x ɛ where the constant in the O1 term is independent of. [α ], [α = ], Self-avoiding walk Self-avoiding walk sometimes abbreviated as SAW is a model for linear polymers. Taking into account the exclusion-volume effect among constituent monomers, we define the SAW two-point function as G p x = ω:o x ω p ω j=1 Dω j ω j 1 s<t 1 δ ωs,ω t, 1.1 where the contribution from the zero-step walk is δ o,x, just as in Notice that the difference between 1.18 and 1.1 is the last product, which is either or 1 depending on whether ω intersects itself or does not. Because of this suppressing factor, the sum called the susceptibility χ p = x Z d G p x 1. is not bigger than x Z d S p x, which is 1 p 1 when p is smaller than the radius of convergence 1, and therefore the critical point p c = sup{p : χ p < } 1.3 must be at least 1. It is known [4] that, if the bubble condition G p c o = x Z d G pc x < 1.4 holds, then χ p p c p 1, 1.5 meaning that the critical exponent for χ p takes on its mean-field value Percolation Percolation is a model for random media. Each bond {u, v} Z d is assigned to be either occupied or vacant, independently of the other bonds. The probability of a bond {u, v} being occupied is defined as pdv u, where p is the percolation parameter. Since 6

7 D is a probability distribution, the expected number of occupied bonds per vertex equals p x o Dx = p1 Do. et G px denote the percolation two-point function, which is the probability that there is a self-avoiding path of occupied bonds from o to x. By convention, G p o = 1. For percolation, the susceptibility χ p in 1. equals the expected number of vertices connected from o. It is known [6] that there is a critical point p c defined as in 1.3 such that χ p is finite if and only if p < p c and diverges as p p c. It is also known that, if the triangle condition G 3 p c o = x Z d G pc x G p c x < 1.6 holds, then χ p diverges in the same way as 1.5. There is another order parameter θ p called the percolation probability, which is the probability of the origin o being connected to infinity. It is known [, 16, 5] that p c in 1.3 can be characterized as inf{p : θ p > } and that, if the triangle condition 1.6 holds, then θ p p p c, 1.7 meaning that the critical exponent for θ p takes on its mean-field value 1, i.e., the value for the survival probability of the branching process The Ising model The Ising model is a model for magnets. et Λ Z d and define the Hamiltonian under the free-boundary condition for a spin configuration ϕ = {ϕ v } v Λ {±1} Λ as H Λ ϕ = J u,v ϕ u ϕ v, 1.8 {u,v} Λ where J u,v = J o,v u is the ferromagnetic coupling and is to satisfy the relation Dx = tanhβj o,x y Z d tanhβj o,y, 1.9 where β is the inverse temperature. et ϕ o ϕ x β,λ = ϕ {±1} Λ ϕ o ϕ x e βhλϕ / ϕ {±1} Λ e βhλϕ. 1.3 Using p = x Z d tanhβj o,x, we define the Ising two-point function G p x as a unique infinite-volume limit of ϕ o ϕ x β,λ : G p x = lim Λ Z d ϕ o ϕ x β,λ

8 It is known [] that the susceptibility χ p defined as in 1. is finite if and only if p < p c and diverges as p p c. It is also known [3, 16] that p c is unique in the sense that the spontaneous magnetization θ p = lim G px 1.3 x also exhibits a phase transition at p c. Unlike the case for percolation, the continuity of θ p in p has been proven for all dimensions, as long as J o,x satisfies a strong symmetry condition called the reflection positivity [4]. Furthermore, it is known [1, 5] that, if the bubble condition 1.4 holds for the critical Ising model, then χ p p c p 1, θ p p p c, 1.33 meaning that the critical exponents for χ p and θ p take on their mean-field values 1 and 1/, respectively The main results et d c = α m, m = { [SAW and Ising], 3 [percolation], 1.34 where m is the number of G pc involved in the bubble/triangle conditions 1.4 and 1.6. In the previous paper [15], we investigated asymptotic behavior of G pc x for α, d > d c and 1 see Theorem 1.7. In the current paper, we investigate the marginal case α =, for which the variance of D diverges logarithmically, and prove the following: Theorem 1.4. et α = and d d c including equality and suppose that D satisfies Assumptions Then there is a model-dependent < such that, for any, G pc x δ o,x + Moreover, there is an ɛ > such that, as x, G pc x = 1 p c γ /v x d log x where the O1 term is independent of. O x d log x O1 log x ɛ, 1.36 Due to the log correction to the standard Newtonian behavior in , we can show that the bubble/triangle conditions hold, even at the critical dimension d = d c. For example, the tail of the sum in the bubble condition 1.4 can be estimated, for any R > 1, as x: x >R G pc x R dr r r 4 d log r,

9 which is finite even when d = 4, due to the log-squared term in the denominator. Also, by the convolution bounds in emma 3.5 below, which is one of the novelties of this paper, we can show that G p c x for d 4 is bounded above by a multiple of x 4 d / log x. Therefore, the tail of the sum in the triangle condition 1.6 can be estimated as x: x >R G pc x G dr p c x R r r 6 d log r, 1.38 which is finite even when d = 6, again due to the log-squared term in the denominator. Therefore: Corollary 1.5. The mean-field results 1.5, 1.7 and 1.33 hold for all three models with α = and sufficiently large, in dimensions d d c including equality. Remark In the previous paper [15], we investigated the other case α and proved the following: Theorem 1.7 Theorems 1. and 3.3 of [15]. et α and d > d c and suppose that D satisfies Assumptions and the following bound on the derivative of D n : for n N and x, y Z d with y 1 3 x, D n x D n x + y + D n x y n Oα y x d+α + Then, there is a model-dependent < such that, for any, G pc x δ o,x + O α. 1.4 x d α As a result, the bubble/triangle conditions 1.4 and 1.6 hold, and therefore the critical exponents for χ p and θ p take on their respective mean-field values. Moreover, there are A = 1 + O 1 {α>} and ɛ > such that, as x, G pc x = A p c γ α /v α 1 + Oɛ x d α x ɛ The extra assumption 1.39 is hard to verify in a general setup. However, we have shown [15] that the compound-zeta distribution 1.15 for α satisfies In fact, as explained in Section 3. see also Remark 3.7, the proof of Theorem 1.4 for α = also works for the case α <, so that we do not have to require 1.39 for α, but not for α >. This is somewhat related to the fact that the multiplicative constant A in 1.41 becomes 1 for α.. The possibility to extend the mean-field results down to d = d c was already hinted in [1, Theorem 1.1], where we have shown that, for d > d c and 1, the Fourier 9

10 transform Ĝpk obeys the following infrared bound, uniformly in k [ π, π] d and p < p c : Ĝ p k = 1 + Oδ m, 1.4 χ 1 p + p1 ˆDk where δ m = [ π,π] d d d k ˆDk π d 1 ˆDk m In fact, we can follow the same line of proof of [1, Theorem 1.1] to obtain 1.4, as long as δ m is sufficiently small. However, for α = and d d c including equality, we have δ m d d k ˆDk k >1/ π d m O m d d k k 1/ k log π k }{{ m } 1.1 = O d Therefore, by taking sufficiently large and using monotonicity in p, we obtain G m p c o = lim G m p p pc o = lim p pc [ π,π] d as long as d d c, hence the mean-field results for all d d c. d d k π d Ĝpk m <, 1.45 Analysis for the underlying random walk In Section.1, we prove Theorem 1.3 for α = the results for α have been proven in [15]. In Section., we complete the proof of 1.1 for α =..1 Proof of Theorem 1.3 The results for α are already proven in [15, Proposition.1]. The proof of 1.19 for α = is easy, as we split the sum at N x 1/ log x 1 and use 1.13 for n N and 1.1 for n N, as follows: S q x δ o,x N D n x + n=1 D n n=n log x O d 1 x d+ 1 log x O d x d+ 1 N n + n=1 1 N + 1 n log n d/ n=n N 1 d/ log N d/ = O d x d 1 log x..1 1

11 et It remains to show 1. for α =. First, we rewrite S 1 x for d > as S 1 x = [ π,π] d d d k π d e ik x 1 ˆDk = Then, for x > 1+η so that δ o,x =, I 1 = = 1.13 dt [ π,π] d µ,, T = x d+ T T T dt [ π,π] d dt e t n= log x d d k e ik x t1 ˆDk dt π d t n n! dt e t δ o,x + which is an error term. Next, we investigate S 1 x I 1. et ω = [ π,π] d n=1 d d k e ik x t1 ˆDk.. π d 1+µ..3 d d k e ik x n ˆDk π d t n n! D n x O log x T = O x d,.4 x d+ log x 1+µ 1 η log Then, we can rewrite S 1 x I 1 as S 1 x I 1 = = T T, 1, R = dt [ π,π] d dt k R d d k e ik x t1 ˆDk dt π d d d k π d e ik x v t k log ω x..5 1 k + 4 I j,.6 j= where I = I 3 = I 4 = T d d k dt e ik x e t1 ˆDk e v t k log 1 k R π d k,.7 R< k 1/ [ π,π] d d d k e ik x T 1 ˆDk π d 1 ˆDk,.8 d d k e ik x T 1 ˆDk π d 1 ˆDk 1 { k >1/}..9 11

12 For I, we first note that, by 1.11, e t1 ˆDk e v t k log 1 k O t k e v t k log 1 k..1 et s = v t k log 1 and r = k R. Since x > 1+η, we have k ds s = 1 dr log 1 r 1 dr log 1 r > 1 dr ωη log r = dr r..11 r R Therefore, for d >, I O O.5 O d.3 = which is an error term because T T R dt t dt t log x O x d dr r rd+ e vtr log v tr log 1 R ds s d+/ T 1 d/ 1 r d+/ s v t log 1 e s R,.1 log x d µ/ d µ.3 > d d + = d + For I 3, since 1.1 holds and log π log π k that I 3 O 1/ R > > for k 1/, there is a c > such dr r rd e c T r = O d T 1 d/ ct c T R ds s sd / e s..14 Since T R as x cf.,.3 and.5, the integral is bounded by a multiple of T R d 4/ e c T R, which is a bound on the incomplete gamma function. Therefore, for N N large enough to ensure N + 4 > d, I 3 O d Rd 4 e c T R Rd 4 O d T T T R N which is an error term. = O +N+4 d1 ω log x 1+µN+1 x d +N+4 d1 ω x > 1+η O log x 1+µN+1,.15 x d +N+4 d1 ωη/1+η 1

13 For I 4, we use 1.9 and a similar argument to.15 to obtain that, for N N large enough to ensure Nη 1/1 + η > d, I 4 O1e T O1 T N which is an error term. So far, we have obtained S 1 x = T ON log x N x N dt k R x > 1+η d d k π d e ik x v t k log O log x N x Nη 1/1+η,.16 1 k + 4 I j..17 To investigate the above integral, we introduce ξ x/ x and change variables as κ = x k. Then, by changing time variables as τ = v t log x, the integral in.6 can be written as x x d d d κ dt T κ x R π exp iκ ξ v t κ log x d x κ = x d d d x κ dτ v log x κ x R π exp log κ iκ ξ τ κ d where M 1 = M = M 3 = = x d v log x v T x x log v T x x log v T x x log v T x x log dτ dτ dτ dτ R d κ > x R κ x R R d d d κ e iκ ξ τ κ π d j=1 log x 3 M j,.18 j=1 d d κ e iκ ξ τ κ,.19 π d d d κ e iκ ξ τ κ,. π d log κ 1 exp τ κ..1 d d κ π d e iκ ξ τ κ log x Notice that the first term in the parentheses in.18 gives the leading term: d d κ dτ e iκ ξ τ κ = dτ e 1/4τ d Γ = = γ.. π d 4πτ d/ 4π d/ R d For M 1, since x /v T log x = v log x µ, we obtain that, for N N large enough to ensure N + 4 > d, M 1 = v T x x log dτ e 1/4τ 4πτ = 1 d/ 4π d/ x /4v T log x x O1 v T log x ds s sd / e s d 4/ e x /4v T log x..3 13

14 Using the exponentially decaying term yields N O1 M 1,.4 log x N+4 dµ/ which gives an error term as long as N + 4 > d. For M, changing the order of integrations and changing variables as r = κ v T log x x yields d d κ M dτ e τ κ π d κ > x R v T x x log d d κ 1 = κ > x R π d κ exp x d / = O1 v T log x = O1 x R d 4 x v T log x κ v T x log x v T R log x dr r rd / e r e v T R log x..5 Using the exponentially decaying term and x > 1+η as in.15.16, we obtain that, for N N large enough to ensure N + 4 > d, x O1log M N+1µ x N+4 d1 ω which gives another error term. x > 1+η For M 3, we first note that, since κ x R = x / 1 ω, 1 exp log κ log κ τ κ log x τ κ exp log x O1log x N+1µ,.6 x N+4 d1 ωη/1+η τ κ log x R log x log κ = τ κ e 1 ωτ κ..7 log x Then, by changing the order of integrations and changing variables as s = ω κ v T we obtain M 3 1 d d κ log x κ x R π d κ log κ dτ τ e ωτ κ = 1 log x κ x R d d κ π d log κ ω κ = O1 x d / log x v T log x log x 1+d µ/ v T x x log log x, x 1 + ω κ v T x log e ω κ v T x x log x ds s log s x ωv T log x sd / 1 + se s. ωv T R log x.8 14

15 Using the triangle inequality log s x ωv T log x we obtain As a result,.3 log s ωv + µ log log x,.9 ωv T R log x ds s log s x ωv T log x sd / 1 + se s ds s log s ωv sd / 1 + se s + Γ d + Γ d µ log log x convergent as long as d > = O1 1 + log log x..3 M 3 O1 log log x log x 1 d µ/ µ< d+ which gives another error term. Summarizing and., we arrive at S 1 x = x d v log x γ x O1 log log log x 3 M j + j=1 4/d+,.31 4 I j,.3 with the error estimates.4,.1,.15.16,.4,.6 and.31. This completes the proof of Theorem 1.3 assuming the properties in Assumptions Proof of the bound 1.1 on D n for α = For n = 1, D = O d is obvious. For n, we recall that D n is bounded as cf., [1, A. and A.4] D n O d 1 j=1 dr r rd e nr π log r + D 1 n..33 Since the second term decays exponentially in n, it suffices to show that 1 dr r rd e nr π log r O n log πn d/..34 et t = n 1/4 so that nt = n. Notice that log π log π > for r 1. By changing r variables as s = nr log π, we have 1 t dr r rd e nr π log r On d/ n log π ds s sd/ e s On d+ 4 e n log π,.35 15

16 which decays much faster than On log πn d/. For the remaining integral over r, t, we change variables as s = nr log π. Then, there is a c > such that r ds s = 1 dr log π r 1 dr log π r cdr r,.36 r t s s s r = n log π n log π = r t n 1 log n + log π 4s n log πn Therefore, t as required. dr r rd e nr π log 4 d/ nt log π t r cn log πn d/ ds s sd/ e s 4d/ Γ d cn log πn,.38 d/ 3 Analysis for the two-point function In this section, we use the lace expansion 1. to prove Theorem 1.4. First, in Section 3.1, we summarize some known facts, including the precise statement of the lace expansion for the two-point function. Then, in Section 3., we prove the infrared bound 1.35 by using convolution bounds on power functions with log corrections emma 3.5 and bounds on the lace-expansion coefficients emma 3.6. The proofs of those two lemmas follow, in Sections , respectively. Finally, in Section 3.5, we prove the asymptotic behavior 1.36 and complete the proof of Theorem ist of known facts The following four propositions hold independently of the value of α >. Proposition 3.1 emma. of [15]. For every x Z d, G p x is nondecreasing and continuous in p < p c for SAW, and in p p c for percolation and the Ising model. The continuity up to p = p c for SAW is also valid if G p x is uniformly bounded in p < p c. Proposition 3. emma.3 of [15]. For every p < p c and x Z d, G p x S p x, pdx G p x δ o,x pd G p x. 3.1 Proposition 3.3 emma.4 of [15]. For every p < p c, there is a K p = K p α, d, < such that, for any x Z d, G p x K p x d α

17 Proposition 3.4 [11] for SAW; [19] for percolation; [8] for the Ising model. There are model-dependent nonnegative functions on Z d, {π p n } n= π p for SAW and {R p n } n=1, such that, for every integer n, G p = { δ + pd + π p n G p + 1 n+1 R p n+1 π n p + π n p pd G p + 1 n+1 R n+1 p [SAW], [percolation & Ising], 3.3 where the spatial variables are omitted e.g., G p for G p x, δ for δ o,x and 1 D = D Doδ, π n p = n j= 1 j π j p. 3.4 Moreover, the remainder term obeys the following bound: R n+1 p { π n+1 p G p [SAW], π n p pd G p [percolation & Ising]. 3.5 Before proceeding to the next subsection, we derive the unified expression 1. from 3.3. To do so, we first assume p < p c and j πj p 1 <, which has been verified for α, d > d c and 1 in [15] and is verified in the next subsection for α =, d d c and 1. Then, by 3.5, we can take the n limit to obtain { δ + pd + π p G p [SAW], G p = 3.6 π p + π p pd G p [percolation & Ising], where π p = lim n π p n. For percolation and the Ising model, if pdo π p 1 < 1 also verified for α, d > d c and 1 in [15], and for α =, d d c and 1 in the next subsection, then G p = π p + π p pd G p pdoπ p G p replace = π p + π p pd G p pdoπ p π p + π p pd G p pdoπ p G p = π p pdoπp + πp pdoπp pd Gp + pdo π p G p replace = π p pdoπ p. + pdo π p + πp pdoπp pd Gp π p + π p pd G p pdoπ p G p = Π p + Π p pd G p, The recursion equation [15, 1.11] is correct for percolation and the Ising model, but not quite for SAW, as long as Do >. To deal with such D, the definition [15, 1.13] of Π p needs slight modification. See 3.1 below. 17

18 where Π p = π p + n=1 pdo nπ n+1 p. 3.8 For SAW, if pdo + π p 1 < 1 also verified for α, d > d c and 1 in [15], and for α =, d d c and 1 in the next subsection, then G p = δ + pd G p + pdoδ + π p Gp replace = δ + pd G p + pdoδ + π p =. δ + pdoδ + π p + + pdoδ + π p Gp replace δ + pd G p + pdoδ + π p Gp δ + pdoδ + π p pd G p = Π p + Π p pd G p, 3.9 where n. Π p = δ + pdoδ + πp 3.1 n=1 3. Proof of the infrared bound 1.35 et α =, d d c and Define λ = sup x o S 1 x x d / log x 1 g p = p sup x o We will show that g p satisfies the following three properties: i g p is continuous and nondecreasing in p [1, p c. ii g 1 1. = O G p x λ x d / log x iii If λ 1 i.e., 1, then g p 3 implies g p for every p 1, p c. Notice that the above properties readily imply G p x λ x d / log x 1 for all x o and p < p c. By Proposition 3.1, we can extend this bound up to p c, which completes the proof of It remains to prove the properties i iii. 18

19 Proof of i. It suffices to show that sup x o G p x/ x d / log x 1 is continuous in p [1, p ] for every fixed p 1, p c. First, by the monotonicity of G p x in p p and using Proposition 3.3, we have G p x x d / log x 1 G p x x d / log x 1 K p x d x d / log x 1 = K p x 4 / log x On the other hand, for any x o with Dx >, there is an R = Rp, x < such that, for all x R, K p x 4 / log x 1 Dx x d / log x Moreover, by using p 1 and the lower bound of the second inequality in 3.1, we have As a result, for any p [1, p ], we obtain sup x o G p x x d / log x 1 = Dx pdx G p x G p x x d log x 1 max x:< x <R G p x x d / log x Since G p x is continuous in p cf., Proposition 3.1 and the maximum of finitely many continuous functions is continuous, we can conclude that g p is continuous in p [1, p ], as required. Proof of ii. as required. g 1 = 1 sup x o By Proposition 3. and the definition 3.11 of λ, we readily obtain G 1 x λ x d / log x 1 1 sup x o S 1 x λ x d / log x 1 1, 3.17 Proof of iii. This is the most involved part among i iii, and here we use the lace expansion. To evaluate the lace-expansion coefficients, we use the following bounds on convolutions of power functions with log corrections, whose proof is deferred to Section 3.5. emma 3.5. For a 1 b 1 > with a 1 + b 1 d, and for a, b with a b when a 1 = b 1, there is an -independent constant C = Cd, a 1, a, b 1, b < such that y Z d a x y 1 y b 1 log x y 1 a log y 1 b C x b 1 log x 1 b d a 1 [a 1 > d], log log x 1 [a 1 = d, a = 1], log x 1 1 a [a 1 = d, a 1], x d a 1 [a 1 < d, a 1 + b 1 > d], x b 1 log x 1 1 a [a 1 < d, a 1 + b 1 = d, a + b > 1]

20 o x o x Figure 1: Examples of the lace-expansion diagrams for SAW and the Ising model left and percolation right. The factor l in 3.19 is the number of disjoint paths in red from o to x using different sets of line segments. Assuming g p 3 and emma 3.5, we prove in Section 3.4 the following bounds on the lace-expansion coefficients {π p n } n= recall that π p for SAW in 3.3. emma 3.6. et cf., 1.34 for the definition of m l = m + 1 m 1 = { 3 [SAW & Ising], [percolation]. Suppose g p 3 and p < p c. Under the same condition as in Theorem 1.4, we have 3.19 pd G p x Oλ x d log x 1 [x Z d ]. 3. Moreover, for SAW, π j p x and for the Ising model and percolation, π j p x O d δ o,x [j = 1], Oλ j+1 x 3 d log x [j ], 1 3 O d j δ o,x + Oλ x l d log x [j =, 1], 1 l x l d Oλ j log x [j ]. 1 l Consequently, we have j πj p 1 < for d d c and 1. Then, by using 3.5 for p < p c, we obtain lim n R p n 1 = and 3.6 with O d δ o,x + Oλ 3 x 3 d log x π p x [SAW], O d 3.3 δ o,x + Oλ x l d log x [Ising & percolation]. 1 l

21 This implies that, for SAW cf., 3.1, pdo δ o,x + π p x O d δ o,x + Oλ 3 x 3 d log x 1, O d hence pdo + π p 1 < 1 for d d c and 1. Also, for the Ising model and percolation cf., 3.8, since d = Oλ for d, pdo π p x O d δ o,x + Oλ 3 x l d log x 1, 3.5 l hence pdo π p 1 < 1 for d d c and 1. We note that, for all three models, l d = d l 1d d c. 3.6 By repeated applications of 3.4 and emma 3.5, Π p x for SAW obeys the bound Π p x δ o,x pdoδ + πp n x n=1 O d n δ o,x + n=1 } {{ } O d n n=1 j=1 O d δ o,x + Oλ 3 x 3 d log x 1 3 n O d n j Oλ 3 j O d 4 j 1 x 3 d j log x 1 3 emma 3.5 n 1 n=1 n O d + Oλ 3 d 4 O d } {{ } O Similarly, by repeated applications of 3.3, 3.5 and emma 3.5, Π p x for the Ising model and percolation obeys the bound Π p x δ o,x π p x δ o,x + π p pdo πp n n=1 } {{ } RHS of 3.7 x O d δ o,x + Oλ x 3 d log x By weakening the Oλ 3 term in the right-most expression of 3.7 to Oλ, Π p x for all three models enjoys the unified bound Π p x δ o,x O d δ o,x + Oλ x l d log x l 1

22 As a result, ˆΠ p 1 O d + Oλ x l d log x = O d, 3.3 x 1 l O l 1d dc and ˆΠ p ˆΠ p k Oλ k x x l d log x Oλ k x 1 l O l 1d dc Now we are back to the proof of iii. First, by summing both sides of 1. over x and solve the resulting equation for χ p, we have χ p = ˆΠ p + ˆΠ p pχ p = ˆΠ p 1 p ˆΠ p. 3.3 Since χ p < because p < p c and ˆΠ p = 1 + O d > for large, we obtain p ˆΠ p, 1, 3.33 which implies p < ˆΠ p 1 = 1 + O d, as required. Next, we investigate G p x. By repeated applications of 1. for N times, we have G p x = Π p x + Π p pd G p x = Π p x + Π p pd Π p x + Π p pd G p x. = N 1 Π p pd Π p x n + Π p pd N G p x n= Notice that, by 3.6, 3.9 and emma 3.5, there are finite constants C, C, C such that Π p Dx 1 C d Dx C λ y y l d log y Dx y l 1 1 C d C λ 3 Dx, 3.35 which is positive for all x, if is large enough see Remark 3.7 below. Therefore, Dx = Π p Dx ˆΠ p 3.36

23 is a probability distribution that satisfies Assumptions see computations below. By this observation, we can take the limit Π p pd N N G p x = p ˆΠp D N G p x, 3.37 N,1 χ p so that G p x = Π p n= p ˆΠp n D n x = Π p S p ˆΠp x, 3.38 where S q is the random-walk Green function generated by the 1-step distribution D with fugacity q [, 1], for which 1.19 holds. By 3.9 and emma 3.5, we obtain that, for x o, G p x 1 + O d S 1 x + δ y,x + Oλ y l d Oλ x y d log y y o 1 l log x y 1 Oλ 4 x d / log x 1 Suppose S 1 x 1 + Oλ 3 S 1 x holds for all x. Then, for x o, G p x 1 + Oλ 3 S 1 x + Oλ 4 x d log x λ x d log x, as required. It remains to show S 1 x 1 + Oλ 3 S 1 x for all x. This is not so hard to verify, as explained now. First, by 3.35 and its opposite inequality with all negative signs replaced by positive signs, Dx Dx 1 = Oλ Also, by and 1.1, Similarly, 1 ˆDk 1 ˆDk 1 ˆDk k log1/ k = 3.36 = 1 + ˆDk ˆΠ p 1+O d 1 ˆDk k log1/ k } {{ } v, 1.4 ˆΠ p ˆΠ p k 1 }{{ ˆDk = 1 + Oλ } Oλ 3 + ˆDk ˆΠ p ˆΠ p ˆΠ p k k log1/ k } {{ }, 3.31 k v Therefore, for large enough, D satisfies all with the same constants as D modulo Oλ 3 terms. Similar analysis can be applied to show that D also satisfies 1.13 with the same constant as D. As a result, we can get S 1 x 1 + Oλ 3 S 1 x for all x. This completes the proof of iii, hence the proof of the infrared bound

24 Remark 3.7. The above proof works as long as α + l 1d d c cf., 3.35, then we can define the probability distribution 3.36 by taking sufficiently large. For short-range models investigated in [17, 18, 8], on the other hand, since α is regarded as an arbitrarily large number, there is no way for 3.35 to be nonnegative for every x. In this case, we may have to introduce a quite delicate function E p,q,r x as in [15, 18] that is required to satisfy some symmetry conditions. Since we do not need such a function for all α and d d c, the analysis explained in this subsection is much easier and more transparent than the previous one in [15, 18]. This is also related to the reason why the multiplicative constant A in the asymptotic expression 1.41 becomes 1 for α. 3.3 Convolution bounds on power functions with log corrections In this subsection, we prove emma 3.5. First, we rewrite the sum in 3.18 as y Z d a x y 1 y b 1 log x y 1 a log y 1 = b y: x y y + = y: x y > y y: x y y x y a 1 y b 1 log x y 1 a log y 1 b x y a 1 log x y 1 a x y a 1 log x y 1 a + x y b 1 log x y 1 b y b 1 log y 1 b y b 1 log y 1 b y a 1 log y 1 a Notice that the ratio of the second term to the first term in the parentheses, which is x y b 1 log x y 1 b x y a 1 log x y 1 a y a 1 log y 1 a y b 1 log y 1 b a1 b x 1 y log x y = a b 1 y log y, is bounded above by an -independent constant C [1, as long as a 1 > b 1, or a 1 = b 1 and a b. Therefore, y Z d a x y 1 y b 1 log x y 1 a log y 1 C b y: x y y x y a 1 y b 1 log x y 1 a log y b Now we consider the following cases separately: a a 1 > d, b a 1 = d, c a 1 < d and a 1 + b 1 d. a et a 1 > d. Since x y y implies y 1 x, and since x 1 x, log x 1 log π log π log x 1,

25 we obtain y: x y y x y a 1 y b 1 log x y 1 a log y 1 b b et a 1 = d. First we split the sum as = y: x y y O1 x b1 log x 1 b y: x y y y 3 x + y: x y y y > 3 x y Z d a x y 1 log x y 1 a } {{ } O d a For the first sum, since x y y implies y 1 x so that 3.47 holds, and since log 3x 1 we obtain y: x y y y 3 x log 3π 4 log π log x 1, x y d log x y 1 a y b 1 log y 1 b O1 x b 1 log x 1 b log log 3x 1 O1 x b1 log x 1 b log log 3π 4 log log π y: x y 3 x 1 [a > 1], log log x 1 [a = 1], log x 1 1 a [a < 1]. log log x 1, 3.5 d x y log x y 1 a For the second sum in 3.49, since y > 3 x implies x y 1 y, and since 3 we obtain y: x y y y > 3 x y y, log y 3 1 log π log 3π x y d log x y 1 a which is smaller than y: x y y y 3 x y b 1 log y 1 b O1 log x 1 a +b 3.51 log y 1, 3.5 y d b1 y: y > 3 } x {{} O1 x b 1, 3.53 c et a 1 < d and a 1 +b 1 d. Similarly to the case b, we split the sum as in 3.49 and evaluate each sum by using 3.47 and 3.5. Then, by discarding the log-dumping term log x y 1 a, the first sum in 3.49 is bounded as x y a 1 y b 1 log x y 1 a log y 1 O1 x b1 b log x b 5 x y a1 y: x y 3 } x {{} O1 x d a 1

26 while the second sum in 3.49 is bounded as y: x y y y > 3 x x y a 1 log x y 1 a O1 x d a 1 b 1 log x 1 a +b y b 1 log y 1 b x y 1 3 y O1 { 1 [a 1 + b 1 > d], y: y > 3 x a y 1 b 1 log y 1 a +b log x 1 [a 1 + b 1 = d, a + b > 1], 3.55 which is smaller resp., larger than 3.54 if a > 1 resp., a < 1. This completes the proof of emma Bounds on the lace-expansion coefficients In this subsection, we prove emma 3.6. Suppose that g p 3 and p < p c. Since G p y = δ o,y + G p y1 {y o} for all three models, we have D G p x = Dx + y o Dx y G p y The first term is easy, because Dx = O x d+ = Oλ log x 1 x 4 1 x d log x 1 For the second term in 3.56, we use g p 3 and emma 3.5 as Oλ x d log x y o Dx y G p y y Z d O x y d+ 3λ y d log y 1 Oλ x d log x This completes the proof of 3.. To prove , we repeatedly apply emma 3.5 to the diagrammatic bounds on π p x in [18, 8]. For example, the lace-expansion diagram in Figure 1 for SAW and the Ising model can be bounded as follows. Suppose for now that each line segment, say, from x to y, represents 3λ x y d x y / log 1, i.e., the assumed bound on the nonzero two-point function. Then, by using emma 3.5 to perform the sum over w, we can show that, for d 4, o y z w x 4 d 1 3λ log π log π C o y z x,

27 where the indicies in the parentheses are summed over Z d. This is due to the following computation: for d 4, w w 3λ y w d 3λ z w d 3λ w x d log y w 1 log z w 1 log w x 1 1 { y w w x } + 1 { y w w x } 1 { z w w x } + 1 { z w w x } 3λ y w d log y w 1 3λ z w d log z w 1 3λ w x d log w x 1 { y w w x } 1 w x 1 y x 1 { z w w x } w x 1 z x + [3 other cases] y x 3λ 3λ d 3λ z x d y w d z w d log y x 1 log z x 1 log y w w 1 log z w +[3 other cases] 1 4 d 1 3λ log π log π C 3λ y x d log y x 1 C y z d y z / log 1 4C 3λ z x d log z x By repeated application of the above inequality, we will end up with o y z w x 4 d 1 3λ log π x 5 log π C 4 d 1 3λ log π log π C o 5 3λ 3 } {{ } Oλ 13 x 3 d log x 1 3, 3.61 which is smaller than , by a factor Oλ 5 for SAW, in particular. This is because, in fact, not every line segment is nonzero. The situation for the Ising model and percolation is harder, because most of the line segments can be zero-length, which do not have small factors of λ. However, the convolution pd G p shows up repeatedly, which has a small factor of λ, as in 3.. This also provides a bound on the main contribution from π p 1 x, as pd G p oδ o,x 3 Do + y o O y d+ This completes the sketch proof of emma λ y d log y δ o,x = O d δ o,x

28 3.5 Proof of the asymptotic behavior 1.36 First we recall Since χ p = ˆΠ p /1 p ˆΠ p diverges as p p c, while ˆΠ p = 1 + O d uniformly in p < p c, we have p c ˆΠpc = 1. Therefore, G pc x = Π pc S 1 x = ˆΠ pc S 1 x + Π pc y S 1 x y S 1 x y o 1/p c The asymptotic expression of S 1 x is the same as that of S 1 x. This can be shown by following the proof of 1. and using the limit To investigate the error term in 3.63, we first split the sum as = y o y:< y 1 3 x + y: x y 1 3 x + y: y x y > 1 3 x For y, since x y 1 x implies x y so that a similar inequality to 3.47 or holds, we have that, for large x, Πpc y S 1 x y S 1 x 3.9 Oλ x l d S1 x y + S log x l 1 x y Similarly, for y Π pc for large x, Πpc y S 1 x y S 1 x 3.9 y y y y: x y 1 3 x y } {{ } Oλ x / log x = Oλ 3 x d l 1d dc log x l Oλ x d log x y: y > 1 3 Oλ y l d log y l x Oλ x l 1d dc /log x l = Oλ 3 x d l 1d dc log x l It remains to investigate y. For that, we first use 1. and the Zd -symmetry of to obtain that, for large x, Πpc y S 1 x y S 1 x y γ Πpc y 1 x + y d x y d v + x d log x + y log x y log x y Oλ Oλ x d + Π pc y log x 1+ɛ y:< y 1 3 } x {{} Oλ 3 8

29 Then, by Taylor s theorem, x ± y d = x d 1 ± x y d/ x + O y x = x d 1 ± d x y x + O y x log x ± y = log x + log x ± y x, 3.68 = log x ± x y x + O y x, 3.69 which implies 1 x + y d log x + y + x y d log x y x d log x O y x d. 3.7 log x Therefore, the first term on the right-hand side of 3.67 is bounded by Oλ x d log x Oλ d l 1d d y c log y y:< y 1 3 x 1 l Oλ This completes the proof of Theorem 1.4. = Oλ 3 x d log x Acknowledgements The work of AS was supported by JSPS KAKENHI Grant Number 18K346. The work of CC was supported by the grant MOST M-4-4 -MY. We are grateful to the Institute of Mathematics and Mathematics Research Promotion Center MRPC of Academia Sinica, as well as the National Center for Theoretical Sciences NCTS at National Taiwan University, for providing us with comfortable working environment in multiple occasions. We would also like to thank the referees for their comments to improve presentation of this paper. References [1] M. Aizenman. Geometric analysis of φ 4 fields and Ising models. Commun. Math. Phys., : [] M. Aizenman and D.J. Barsky. Sharpness of the phase transition in percolation models. Commun. Math. Phys., : [3] M. Aizenman, D.J. Barsky and R. Fernández. The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys., :

30 [4] M. Aizenman, H. Duminil-Copin and V. Sidoravicius. Random currents and continuity of Ising models spontaneous magnetization. Commun. Math. Phys., : [5] M. Aizenman and R. Fernández. On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys., : [6] M. Aizenman and C.M. Newman. Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys., : [7] M.C. Angelini, G. Parisi and F. Ricci-Tersenghi. Relation between short-range and long-range Ising models. Phys. Rev. E, 89 14: 61. [8] A. Bendikova, W. Cygana and B. Trojanb. imit theorems for random walks. Stoch. Proc. Appl., 17 17: [9] A. Bendikova and W. Cygana. α-stable random walk has massive thorns. Colloq. Math., : [1] E. Brezin, G. Parisi and F. Ricci-Tersenghi. The crossover region between long-range and short-range interactions for the critical exponents. J. Stat. Phys : [11] D. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Comm. Math. Phys : [1].-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Relat. Fields 14 8: [13].-C. Chen and A. Sakai. Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation. Probab. Theory Relat. Fields 145 9: [14].-C. Chen and A. Sakai. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Probab : [15].-C. Chen and A. Sakai. Critical two-point functions for long-range statisticalmechanical models in high dimensions. Ann. Probab : [16] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys : [17] T. Hara. Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36 8:

31 [18] T. Hara, R. van der Hofstad and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 3: [19] T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys : [] T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys : [1] M. Heydenreich, R. van der Hofstad and A. Sakai. Mean-field behavior for longand finite-range Ising model, percolation and self-avoiding walk. J. Stat. Phys. 13 8: [] J. ebowitz. GHS and other inequalities. Comm. Math. Phys : [3] M. ohmann, G. Slade and B.C. Wallace. Critical two-point function for long-range On models below the upper critical dimension. J. Stat. Phys : [4] N. Madras and G. Slade. The Self-Avoiding Walk Birkhäuser, [5] M.V. Menshikov. Coincidence of critical points in percolation problems. Soviet Mathematics, Doklady : [6] B.G. Nguyen and W-S. Yang. Triangle condition for oriented percolation in high dimensions. Ann. Prob : [7] A. Sakai. Mean-field critical behavior for the contact process. J. Stat. Phys. 14 1: [8] A. Sakai. ace expansion for the Ising model. Commun. Math. Phys. 7 7: [9] A. Sakai. Application of the lace expansion to the ϕ 4 model. Commun. Math. Phys :