RIMS Kôkyûroku Bessatsu
B79 (2020), 51–62

Crossover phenomena in the critical behavior for
long-range models with power-law couplings
By

Akira SAKAI∗

Abstract
This is a short review of the two papers [9, 10] on the x-space asymptotics of the critical
two-point function Gpc (x) for the long-range models of self-avoiding walk, percolation and theIsing model on Zd , defined by the translation-invariant power-law step-distribution/couplingD(x) ∝ |x|−d−α for some α > 0. Let S1 (x) be the random-walk Green function generated by
D. We have shown that
• S1 (x) changes its asymptotic behavior from Newton (α > 2) to Riesz (α < 2), with log
correction at α = 2;
• Gpc (x) ∼ pAc S1 (x) as |x| → ∞ in dimensions higher than (or equal to, if α = 2) the
upper critical dimension dc (with suﬃciently large spread-out parameter L). The modeldependent A and dc exhibit crossover at α = 2.
The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp
asymptotics of S1 , (ii) bounds on convolutions of power functions (with log corrections, if α = 2)
to optimally control the lace-expansion coeﬃcients πp(n) , and (iii) probabilistic interpretation(valid only when α ≤ 2) of the convolution of D and a function Πp of the alternating series∑∞n (n)n=0 (−1) πp . We outline the proof, emphasizing the above key elements for percolation inparticular.

§ 1.

Introduction and the main results

Since the dawn of research on phase transitions and critical behavior, it has beenstandard to investigate short-range models, among which the nearest-neighbor modelon Zd is the most popular. Thanks to intensive studies for more than half a century,
Received December 26, 2018. Revised October 15, 2019.
2010 Mathematics Subject Classification(s): 60K35, 82B20, 82B27, 82B41, 82B43.
Key Words: Random walk, self-avoiding walk, percolation, Ising model, power-law coupling, critical two-point function, critical dimension, crossover phenomena, lace expansion.
Supported by JSPS KAKENHI Grant Number 18K03406.
∗ Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan.
https://orcid.org/0000-0003-0943-7842
c 2020 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
⃝

52

Akira Sakai

nearest-neighbor bond percolation is now known to exhibit a phase transition for alld ≥ 2 and mean-field behavior (i.e., the critical two-point function Gpc (x) decays as|x|2−ηshort −d with the mean-field value ηshort = 0) for all d ≥ 11 [11, 12]. Believing
in universality, we expect the mean-field behavior for all dimensions above the uppercritical dimension dshort = 6 for short-range percolation [15].
Recently, long-range random walk and statistical-mechanical models defined by thepower-law step-distribution/coupling D(x) ∝ |x|−d−α , α > 0, have regained popularity,
due to unconventional macroscopic behavior [4,6–10,18,19]. Among those references, aninfrared bound and mean-field behavior are proven for long-range oriented percolation(OP for short) on Zd>dc ×Z+ [6,7] and for long-range models of percolation, self-avoiding
walk (SAW for short) and the Ising model on Zd>dc [18], where2[OP, SAW & Ising],(1.1)
dc = (α ∧ 2) ×
3[percolation].
Also, an asymptotic expression of the gyration radius for long-range models of SAW
and OP for d > dc are proven in [8]. In physics, Brezin, Parisi and Ricci-Tersenghi[4] conjectured that Gpc (x) would decay as |x|α∧(2−ηshort )−d if α ̸= 2 − ηshort , and as|x|α−d / log |x| if α = 2 − ηshort . We have shown in [9, 10] that the conjectured behavior
holds true for d > dshort (= dc with α = ∞), because ηshort = 0, with suﬃciently large
spread-out parameter L [13, 14, 23]. In fact, the obtained results are much stronger, assummarized as follows.
Theorem 1.1 (Proposition 2.1 of [9] and Theorem 1.3 of [10]).
−d−α, i.e.,
1 and D(x) ≍ L1d ( |x|L ∨ 1)(1.2)∃

c > 0, ∀ x ∈ Zd , ∀ L ∈ [1, ∞) : c ≤

Let

(1.3)

Γ( d−α∧2 )γα = α∧2 d/22 α∧2 ,2π Γ( 2 )

D(x)
1 |x|
( ∨ 1)−d−α
Ld L

≤


1 − D̂(k) lim|k|→0 |k|α∧2vα =
1 − D̂(k) lim|k|→0 |k|2 log(1/|k|)Let α > 0, L ≥1.
c[α ̸= 2],[α = 2],∑
ik·x
where D̂(k) =D(x). Then, for all d > α ∧ 2, the random-walk Greenx∈Zd efunction S1 (x) generated by the step distribution D exhibits the following asymptotic
behavior: there is an ϵ > 0 such that, as |x| → ∞,)(O(Lϵ ) 1+[α ̸= 2],
|x|ϵγα /vα×(1.4)
S1 (x) =)(|x|d−α∧2 
1
O(1)



[α = 2],
1+
log |x|(log |x|)ϵCrossover phenomena in the critical behavior for long-range models with power-law couplings53

where the O(1) term is independent of L.
Theorem 1.2 (Theorem 1.2 of [9] and Theorem 1.6 of [10]).
Let D be the sameas in Theorem 1.1 and recall the definition (1.1) of dc . Suppose d > dc for α ̸= 2 orthat d ≥ dc for α = 2. For α > 2, we also assume a bound on the “derivative” of D (seethe last part of Section 3). Then, there is an L0 (d) < ∞ such that, for any L ≥ L0 ,there are A = 1 + O(L−2 )1{α>2} and ϵ > 0 such that, as |x| → ∞,

(1.5)

)
(
O(Lϵ )
 1 + |x|ϵA γα /vα
Gpc (x) =×()
pc |x|d−α∧2 
O(1)
1



1+
log |x|(log |x|)ϵ[α ̸= 2],[α = 2],where the O(1) term is independent of L.
In short, the critical two-point function Gpc (x) exhibits the same asymptotic behavior as S1 (x), modulo multiplication of the model-dependent constant A/pc , for alld > dc (with large spread-out parameter L) and, most interestingly, for d = dc when
α = 2. For d ∈ (dc , dshort ), which is not empty for α < 2 and in which ηshort is believed
to be nonzero, Theorem 1.2 claims that Gpc (x) decays as |x|α−d , not as |x|2−ηshort −d .
This power-law behavior has been extended even below dc by Lohmann, Slade andWallace [19] using a rigorous version of the ε-expansion.
§ 2.

Key ideas for the proof of Theorem 1.1

Let D∗n be the n-fold convolution of D (i.e., the n-step distribution) and denote
by Sq the random-walk Green function generated by D with survival rate q ∈ [0, 1]:(2.1)

D∗n (x) = (D∗(n−1) ∗ D)(x) ≡∑D∗(n−1) (y) D(x − y),

y

(2.2)

Sq (x) =∞∑q n D∗n (x).

n=0

Let
(2.3)

|||x|||r =π(|x| ∨ r)2[x ∈ Rd , 1 ≤ r < ∞],where | · | is the Euclidean norm. Suppose that, as explained in (1.2), D(x) decays as(2.4)

x −d−αD(x) ≍ L−d ||| L|||1≡ Lα |||x|||−d−α
.
L

54

Akira Sakai

An example of D is the following compound zeta distribution [9]:(2.5)D(x) =∑t∈N

UL∗t (x)t−1−α/2
ζ(1 + α/2)[x ∈ Zd ],where UL is the uniform distribution over the d-dimensional box of side-length 2L.
The step distribution D in (2.4) satisfies the following properties (D1)–(D3) thatare essential to the proof of (1.4).
(D1) k-space bounds [6, Proposition 1.1] (and [10, Assumption 1.1]): ∃ ∆ = ∆(L) ∈ (0, 1)
such that< 2 − ∆[∀ k ∈ [−π, π]d ],(2.6)
1 − D̂(k)> ∆[|k| > 1/L],
and for |k| ≤ 1/L,
1 − D̂(k) ≍ (L|k|)α∧2 ×

(2.7)


1
log

[α ̸= 2],π2L|k|[α = 2].

(D2) k-space asymptotics [8, Lemma A.1] (and [10, Assumption 1.1]): ∃ ϵ > 0 such that,as |k| → 0,(
 1 + O(Lϵ |k|ϵ ))[α ̸= 2],α∧2
(2.8)
1 − D̂(k) = vα |k|× ()
 log 1 + O(1)[α = 2],L|k|where the constant in the O(1) term is independent of L.
(D3) x-space bounds [9, (1.19)–(1.21)] (and [10, Assumption 1.2]): ∀ n ∈ N and ∀ x ∈ Zd ,n−d/(α∧2)[α ̸= 2],
(2.9)
∥D∗n ∥∞ ≤ O(L−d ) ×(n log πn )−d/2[α = 2],21[α ̸= 2],O(Lα∧2 )(2.10)
D∗n (x) ≤ n×d+α∧2
log ||| x |||1|||x|||L[α = 2].
L

For example, to show (2.7) for |k| ≤ 1/L, we first split the sum as(2.11)
1 − D̂(k) ≍ L

α∑x|||x|||−d−α(1 − cos k · x)L(

1{|x|<L} + 1π}{L≤|x|≤ 2|k|+1)
π{|x|> 2|k|}.

Crossover phenomena in the critical behavior for long-range models with power-law couplings55

It is easy to see that the contributions from the first and third indicators are O(L2 |k|2 )and O(Lα |k|α ), respectively. The contribution from the second indicator is the main
term since
∑

Lα

∑

|||x|||−d−α(1 − cos k · x) ≍ Lα |k|2
Lπ
L≤|x|≤ 2|k||x|−d−α+2πL≤|x|≤ 2|k|(L|k|)α∧2≍
(L|k|)2 log(2.12)[α ̸= 2],π2L|k|[α = 2].

To prove (1.4), we first rewrite S1 (x) for the transient case d > α ∧ 2 as∫dd k
e−ik·x=
S1 (x) =d[−π,π]d (2π) 1 − D̂(k)∫ ∞∫dd k −ik·x−t(1−D̂(k))
ed0[−π,π]d (2π)∫ ∞ ∫dd k −ik·x−t(1−D̂(k))=
dte+ E1 ,
d0|k|≤R (2π)(2.13)

dt

where R is arbitrary for the moment. Then, by replacing 1 − D̂(k) by its limit (2.8), we
can further rewrite S1 (x) for α ̸= 2 as
∫ ∞(2.14)

S1 (x) =0∫

dd k −ik·x−vα t|k|α∧2e+ E1 + E2 ,dt
dRd (2π)and for α = 2 as
∫ ∞(2.15)

S1 (x) =

∫
dt

0

1
dd k −ik·x−v2 t|k|2 log L|k|e
+ E1 + E2 .
d
Rd (2π)Since∫ ∞(2.16)dt e
0−vα t|k|α∧2

1
1==
vα |k|α∧2vα Γ( α∧2
2 )

∫ ∞
0

dt (α∧2)/2 −t|k|2te,t

we readily obtain for α ̸= 2 that
(2.17)1
S1 (x) − E1 − E2 =vα Γ( α∧2
2 )

∫ ∞
0

dt (α∧2)/2tt∫

dd k −ik·x−t|k|2
γα /vαe=.
d|x|d−α∧2Rd (2π){z}|= (4πt)−d/2 exp(−|x|2 /(4t))

Using the k-space and x-space bounds (D1) and (D3) and choosing R accordingly (asin [9, (2.20)]), we can show that E1 + E2 is the error term in (1.4). See [9, Section 2.1]
for more details.

56

Akira Sakai|x|v2 tFor α = 2, we change variables as ξ = x/|x|, κ = |x|k and τ = |x|
2 log L to obtain

(
)
dd κ
v2 t|κ|2|x|
S1 (x) − E1 − E2 = |x|
exp − iκ · ξ −
log
dt
d
|x|2
L|κ|
0
Rd (2π)
(
|x| )
∫
∫
∞
log
dd κ
|x|2−dL|κ|=dτexp − iκ · ξ − τ |κ|2
d
|x|
(2π)
d
v2 log L 0
log |x|RL∫d2−d ∫ ∞|x|d κ −iκ·ξ−τ |κ|2
(2.18)
=
dτ
e
+E3 .
d
|x|
v2 log L | 0
Rd (2π)
{z
}
−d

∫ ∞

∫

= γ2

Again, by using the k-space and x-space bounds on D and choosing R accordingly (asin [10, (2.5)]), we can show that E1 + E2 + E3 is the error term in (1.4). See [10,Section 2.1] for more details. This completes the sketch proof of Theorem 1.1.
§ 3.

Key ideas for the proof of Theorem 1.2

The proof of Theorem 1.2 is based on the lace expansion, which is one of the few
methods to prove mean-field results mathematically rigorously. Since its invention by
Brydges and Spencer for weakly SAW [5], the method has been extended to strictly
SAW [17], oriented/unoriented percolation [15,21], lattice trees and lattice animals [16],the contact process [22], the Ising and φ4 models [23, 24].
The lace expansion yields a formal recursion equation for the two-point function
Gp (x), which is similar to the recursion equation for the random-walk Green function
Sp (x). For (strictly) SAW, Gp (x) is defined as(3.1)

Gp (x) =

∑
ω:o→x

|ω|

p

|ω|
∏
j=1

D(ωj − ωj−1 )∏(1 − δωs ,ωt ),s<twhere the sum is over the paths ω from o to x. The contribution from the zero-stepwalk is regarded as δo,x . The last product over s, t is either 0 or 1 depending on whether
or not ω intersects to itself.
For Bernoulli bond percolation, in which each bond {u, v} is occupied with probability pD(v − u) independently of the other bonds, the two-point function is defined
as
(3.2)

Gp (x) = Pp (o ←→ x),

where Pp is the induced law from the above bond-occupation probability (p(1 − D(o))
is the expected number of occupied bonds per vertex), and {o ←→ x} is the event thateither x = o or there is a self-avoiding path of occupied bonds from o to x.

Crossover phenomena in the critical behavior for long-range models with power-law couplings57

For the Ising model, see, e.g., [10, Section 1.2.4].
Due to monotonicity in p and subadditivity in self-avoiding paths, the critical point
pc is characterized by the divergence of the susceptibility χp for all models, as follows:
∑
(3.3)
χp =
Gp (x),
pc = sup{p ≥ 0 : χp < ∞}.
x

The proof of Theorem 1.2 consists of the following two steps:x
Step 1: Prove that Gp (x) is bounded by 2λ|||x|||α∧2−dif α ̸= 2 and by 2λ|||x|||2−dL
L / log ||| L |||1
if α = 2, uniformly in x ∈ Zd and p < pc , where

d−α∧2

[α ̸= 2],
sup S1 (x)|||x|||L
x̸=o
(3.4)
λ=
x

log ||| L|||1[α = 2],
sup S1 (x)|||x|||d−2Lx̸=owhich is of order L−α∧2 , by Theorem 1.1.
Step 2: Use the lace expansion as a recursion equation for Gpc (x) to derive its asymptotic expression.
To complete Step 2 is rather straightforward as soon as Step 1 is completed;see [9, Section 3.3] for α ̸= 2 and [10, Section 3.5] for α = 2. To complete Step 1, itsuﬃces to show that gp , defined asGp (x)[α ̸= 2],p ∨ supα∧2−dx̸=o λ|||x|||L
gp =
(3.5)
Gp (x)
[α = 2],p ∨ sup2−dxx̸=o λ|||x|||L / log ||| L |||1
satisfies the following three properties:(S1.1) g1 ≤ 1.
(S1.2) gp is continuous (and nondecreasing) in p ∈ [1, pc ).
(S1.3) gp ≤ 3 implies gp ≤ 2 for every p ∈ (1, pc ), if λ ≪ 1.
The third property implies that there is a prohibited region in the p–gp plane. Therefore,
gp is either ≤ 2 or > 3, as long as p ∈ (1, pc ). However, due to the continuity (S1.2)
with the initial condition (S1.1), the possibility of gp > 3 is eliminated. This completes
Step 1.
(S1.1)–(S1.2) are not so diﬃcult, due to [10, Propositions 3.1–3.3]. To show (S1.3),
we use the lace expansion, which is formally written as(3.6)

Gp (x) = Πp (x) + (Πp ∗ pD ∗ Gp )(x),58

Akira Sakai

where (cf., [10, Section 3.1])

∞
∑
(
)∗n


δ
+
− pD(o)δ + πp (x)
 o,x

(3.7)

Πp (x) =[SAW],

n=1

∞
∑
(
)n



− pD(o) πp∗(n+1) (x)πp (x) +[Ising & percolation].

n=1

Here, πp is the alternating series of the nonnegative lace-expansion coeﬃcients {πp(n) }∞n=0(0)(πp ≡ 0 for SAW):(3.8)

πp (x) =n∑(−1)n πp(n) (x).

n=0

The proof of Item (S1.3) goes as follows.
(i) Bound πp(n) in terms of Gp by using correlation inequalities, such as the BK inequality for percolation [3].
(ii) Derive an optimal x-space bound on Πp in (3.7) by applying the hypothesis gp ≤ 3
to the bounds on πp(n) obtained in (i) and using convolution bounds (see below) on
power functions, with log corrections for α = 2.
(iii) Prove the improved bound gp ≤ 2 by applying the bound on Πp obtained in (ii)to (3.6).
From now on, we restrict our attention to percolation. By the BK inequality, thefirst few terms are bounded as(3.9)πp(0) (x) ≤ Gp (x)2 ,πp(1) (x) ≤ ox,πp(2) (x) ≤ ox + ··· ,where each line segment represents Gp , small filled rectangles are pD and unlabeled
vertices are summed over Zd . For more explanation on those diagrammatic expressions,
we refer to the original paper [15]. Then, we use gp ≤ 3 and the following convolution
bounds:Lemma 3.1 (Lemma 3.5 of [10]).
For a1 ≥ b1 > 0 with a1 + b1 ≥ d, andfor a2 , b2 ≥ 0 with a2 ≥ b2 when a1 = b1 , there is an L-independent constant C =Crossover phenomena in the critical behavior for long-range models with power-law couplings59

C(d, a1 , a2 , b1 , b2 ) < ∞ such that
(3.10)∑−b1
1|||x − y|||−a|||y|||LL
a2 (log ||| y ||| )b2(log ||| x−y
L 1
L |||1 )
y∈Zd


Ld−a1





x

|||1
log log ||| L−b1
C |||x|||Lx≤
× (log ||| L|||1 )0∨(1−a2 )x(log ||| L|||1 )b2 
1|||x|||d−a

L



|||x|||b1 (log ||| x ||| )0∨(1−a2 )L 1L[a1 > d],[a1 = d, a2 = 1],[a1 = d, a2 ̸= 1],[a1 < d, a1 + b1 > d],[a1 < d, a1 + b1 = d, a2 + b2 > 1].

Take πp(1) (x) for α = 2, for example. By repeated applications of the above convolution bounds, we can reduce the number of vertices (and line segments) one by one, as
depicted as follows:
gp ≤3
d≥4

(3.11)

o

x

≲

gp ≤3
d≥6

o

x + o

x

≲

o

x.

Explanation of the above inequality. Let v be the unlabeled top-right vertex in
the leftmost figure at which three line segments (each in red, blue and black) meet, andlet y, z be the other end vertices of the horizontal (in red) and vertical (in black) line
segments, respectively. In the first inequality, we use (3.10) between the vertical line
segment and one of the other two line segments, depending on whether |x − v| ≥ |y − v|
or |x − v| ≤ |y − v|. If |x − v| ≤ |y − v|, then |x − y| ≤ |x − v| + |y − v| ≤ 2|y − v| andtherefore∑
v:|x−v|≤|y−v|(3.12)≤|||y − v|||2−d|||z − v|||2−d|||x − v|||2−d
L
L
L
x−v
y−v
z−v
log ||| L |||1 log ||| L |||1 log ||| L |||1

2−d ∑
2−d
d≥4
||| x−y|||x − z|||4−d|||x − v|||2−d|||z − v|||2−d∃ ′ |||x − y|||L
2 |||L
L
L
L
≤
C
,
x−y
x−v
z−v
x−y
x−z
log
|||
|||
log
|||
|||
log
|||
log ||| 2L |||1 v
log ||| L |||1
1
1
L
L
L |||1|{z}
blue-dottedwhich is depicted as the left figure in the middle expression in (3.11). Then, by gathering
all line segments meeting at z (denote the other end vertex of the horizontal line segmentby u) and using (3.10) again, we obtain(3.13)∑ |||x − z|||(4−d)+(2−d) |||u − z|||2−d d≥6 |||x − u|||2−d
L
L
L
≤ C
,
x−z
u−z
x−u
2
(log ||| L |||1 )log ||| L |||1
log ||| L |||1z

60

Akira Sakai

which yields the rightmost figure of (3.11). We should emphasize that the above boundholds even at dc = 6, because of the log-squared term in the denominator. This is oneof the reasons why the mean-field results1 hold for d ≥ dc (including equality) when
α = 2.
The other case |x − v| ≥ |y − v| can be evaluated similarly, and we refrain fromshowing it here.
Applying the same analysis to the other πp(n) and using (3.7)–(3.8), we can get(cf., [9, (3.4)] and [10, (3.29)])
|||x|||(α∧2−d)ℓ[α ̸= 2],L−d2
(3.15) |Πp (x) − δo,x | ≤ O(L )δo,x + O(λ ) ×
(|||x|||2−d / log ||| x |||1 )ℓ[α = 2],
L

L

where
(3.16)ℓ=2[percolation],

3[SAW & Ising].

Notice from (3.15) that, if α < 2 and d > dc or if α = 2 and d ≥ dc , then Πp ∗ D in(3.6) can be treated, after normalization, as a probability distribution. For α = 2, forexample, there are finite constants c, c′ , c′′ such that(Πp ∗ D)(x)(3.17)∑ |||y|||ℓ(2−d)
L
D(x − y)
y
ℓ
|||
)
(log
|||
1
L
y

(3.15)≥(1 − cL−d )D(x) − c′ λ2Lemma 3.1

(1 − cL−d − c′′ λ3 )D(x),≥which is positive for all x, if λ ≪ 1. Therefore,
D(x) =(3.18)(Πp ∗ D)(x)Π̂p (0)is a probability distribution that satisfies all the properties in (D1)–(D3), and its Green∑∞function n=0 D∗n (x) is bounded by (1 + O(λ3 ))S1 (x) for every x (see [10, Section 3.2]
1 The bubble condition G∗2 (o) < ∞ for SAW/the Ising model and the triangle condition G∗3 (o) < ∞pc
pc
for percolation are suﬃcient conditions for the susceptibility χp and other observables to exhibit

their mean-field behavior. The log correction for α = 2 is the key to extend the mean-field results
down to d = dc since, for example, the tail of the sum in the triangle condition can be estimated,
for any R > 1, asd≥4 ∫ ∞ dr∑
r6−d d≥6(3.14)(x)≲< ∞.
Gpc (x) G∗2pcr (log r)2
R
x:|x|>RCrossover phenomena in the critical behavior for long-range models with power-law couplings61

for more details). By (3.15) and Lemma 3.1, we obtain that, for x ̸= o,(

2−d)
4 |||x|||L
Gp (x) ≤ 1 + O(λ ) (Πp ∗ S1 )(x) ≤ 1 + O(λ ) S1 (x) + O(λ )xlog ||| L
|||1
3

)

(

3

|||x|||2−d
L
≤ 2λ
,
x
log ||| L
|||1

λ≪1

(3.19)

as required. This completes all the steps (i)–(iii) for α ≤ 2.
If α > 2, then we can no longer interpret Πp ∗ D as a probability distribution,
because the second term in (3.17) decays slower than D; this is why the model-dependent
multiplicative constant A in (1.5) is reduced to 1 only when α ≤ 2. To overcome thisdiﬃculty for α > 2, we assume that the “derivative” of the n-step distribution D∗nobeys the following bound: for |y| ≤ 31 |x|,(3.20)

D∗n (x) −O(Lα∧2 ) |||y|||2L
D∗n (x + y) + D∗n (x − y).
≤nd+α∧2+22|||x|||LWe have shown in [9] that the compound zeta distribution (2.5) for α ̸= 2 satisfies theabove assumption. See [9, Appendix] for more details.
Acknowledgements
This work was supported in part by JSPS KAKENHI Grant Number 18K03406,
and in part by JST CREST Grant Number JP22180021. I would like to thank LungChi Chen for the long-time collaboration [6–10] on the long-range models defined by
the power-law decaying step-distribution/coupling. I would also like to thank the organizers of 17th International Symposium “Stochastic Analysis on Large-scale Interacting
Systems” at RIMS, Kyoto University, during November 5–8, 2018, for the opportunity
to speak on the topic of this article. Finally, I would like to express my gratitude to thereferee for positive feedback and valuable suggestions to improve presentation of thispaper.

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