Boundary effects in a two-dimensional Abelian sandpile

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Boundary effects in a two-dimensional Abelian sandpile

J. Brankov, E. Ivashkevich, V. Priezzhev

To cite this version:

J. Brankov, E. Ivashkevich, V. Priezzhev. Boundary effects in a two-dimensional Abelian sandpile.

Journal de Physique I, EDP Sciences, 1993, 3 (8), pp.1729-1740. �10.1051/jp1:1993212�. �jpa-00246828�

J.

Phys.

I France 3 (1993) 1729-1740 AUGUST 1993, PAGE 1729

Classification Physic-s Abstracts

05.40 05.60 46. lo 64.60

Boundary effects in

_a

two-dimensional Abelian sandpile

J. G. Brankov

(*),

E. V. Ivashkevich and V. B. Priezzhev

Laboratory

of Theoretical

Physics,

Joint Institute for Nuclear Research, Dubna, 141980, Russia

(Receii'ed 22 Ma;ch J993, accepted 25

April

J993)

Abstract. We study boundary and finite-size effects in the Abelian

sandpile

model due _to Bak, Tang and Wiesenfeld. In the _case of

half-plane

geometry, ^the

probability

iPj (r) of _a unit height at the

boundary,

and at a distance _r inside the sample is found for open and closed boundary

conditions. The

leading asymptotic

form of the correlation functions for the unit

heights, tPj,

(r), in the

strip

and

half-plane geometries

is obtained for different boundary conditions too.

Our results confirm the

hypothesis

that the unit

height

behaves like the local energy operator ^{in the} zero-component ^{limit of the Potts model.}

1. Introduction.

Since the

publication

of the works

by

Bak,

Tang

and Wiesenfeld I], critical Abelian

sandpile

models _are in the focus of

comprehensive studies,

_{see, e-g-}

[2, 3,

4,

5].

The stochastic

evolution of _a

sandpile naturally

leads to a state of

self-organized criticality (SOC),

which is characterized

by

correlations with

power-law decay

in space and time. As the

spatial _aspects

of the SOC _are similar _to those of _a critical _state in statistical

mechanics,

the program for

studying

the structure of SOC states

essentially parallels

the _one for the usual statistical models like the

Ising

_or the Potts models. Besides the evaluation of bulk critical exponents, an

important

item in this program is the deterrnination of different surface exponents

describing

the behaviour of correlations _{near a}

boundary.

In the two-dimensional _case, finite-size

analysis [6]

and conformal

field-theory [7]

establish

relationships

between surface and bulk

properties

of the models, which lead _{to a}

complete description

^of the

spatial

structure at

criticality. Thus, finding

a

correspondence

between the observables of _a

sandpile

and _an

appropriate

model in

statistical mechanics at different

boundary

conditions _{seems an} attractive task.

The number of distinct recurrent

configurations

of the Abelian

sandpile

model _{on an}

arbitrary

lattice has been

expressed through

the number of

spanning

trees on the _same lattice

[8, 9].

The statistics of

spanning

_trees

obeys

the Kirchhoff theorem and _can

be,

in turn, related

(*) On leave of absence from Institute of Mechanics and Biomechanics, Bulgarian

Academy

of Sciences, lll3Sofia, Bulgaria.

to the _q

=

0 limit of the q-component ^{Potts model}

[10, 11].

In

drawing

a further

analogy,

however, one encounters the

difficulty

of

identifying

local

sandpile

observables _as site variables in the Potts model. The natural formulation of _a

sandpile

model is

given

in _terms of

integer height

variables z, at each lattice site I efif and

toppling

rules. The stable

configurations

in the SOC _state of the Abelian model _{are sets} of

heights (z,,

I _e

fit),

where

z~ e

(1, 2, 3, 4). Majumdar

and Dhar

[12]

have calculated the

probability iJ'j

of

finding

the value z, =

I and the correlation function

Pi

j

(r)

for two sites of unit

height

at a distance _r apart.

In the _two dimensional _case, the result

iJ°j (r)

r~

~',

_with.<

= 2, has been obtained for

large

r, which

provided grounds

for

suggesting [9]

that

iJ°jj(r)

is the counterpart ^{of the} energy- energy correlator in the zero-component ^{Potts model. For the Potts}

model,

besides the bulk exponent x = 2, the surface correlations

decay

exponent ^{is known} too, _{xii =} 2, in the _case of free

boundary

conditions

[13, 14].

The value of the surface exponent xii, in tum, is related _to the

amplitude

A of the inverse correlation

length

f~ ^'

= AIL, which controls the

exponential decay

of

pair

correlations

along

_an

infinitely long strip

of width L.

Namely,

the

amplitude-

exponent ^relations

[15]

~

~~~j

,

~~~d~c

b-c.

follow from finite-size

scaling [6]

and the conformal

properties

of the model

[14].

In this paper, we consider the behaviour of the

probability iJ°j

and the correlation function

£Pjj (r)

_near the

boundary

of the Abelian two-dimensional

sandpile

^model

[8].

The

toppling

rules _are

specified by

^{the matrix} A with elements A,~ for sites I,

j

in the bulk of the lattice

given by

~4,

^{I =}

^j

A,~ ⁼

I, ii j

=

(2)

0

,

otherwise

Here

ii j

denotes the distance between sites and

j.

To formulate the

toppling

rules at the

boundary

3fif of the

sandpile,

_{we use} three standard

boundary

value

problems

for the

Laplacian

_{on a} finite lattice fit.

I.

Open boundary

conditions

(or

Dirichlet

boundary conditions),

when

A,,

=4 for I _e

3fif, and, therefore,

the sand

particles

_are allowed _to leave the system

through

^the

boundary.

2. Closed

boundary

conditions (or Neumann

boundary conditions),

when

j,

₌ ^{3 for}

I e

3fif,

and the sand

panicles

cannot leave the system

through

3fif.

3. Periodic

boundary

conditions, when A,~ =

4 _as in _case I, but _now the lattice is

wrapped

on a torus and

ii j

^is the distance

along

the surface of the _torus.

Obviously,

if all the boundaries _are closed _or

periodic

_no

steady

state of the

sandpile

is

possible. Therefore,

in

considering

the

strip

_{geometry, we} start with _a

rectangular shape

of the

lattice

fit, impose

_open

boundary

conditions _on the vertical

edges

of the

rectangle

and

boundary

conditions of _one of the types 1, ² or 3 _on the

opposite

horizontal

edges.

Then, we let the horizontal size of the

rectangle

tend to

infinity.

The

following

results _are

reported

here.

1. I IN THE CASE OF HALF-PLANE GEOMETRY. The

probability iJ'j (0

) of _a

height

_z

= I at the

boundary

is obtained _: for open

boundary

conditions

iJ'j (0)

=

~ ~~

+

~~~ ~~~

= 0.10382

,

(3)

2 _w 3

w 9 _w

N° 8 BOUNDARY EFFECTS IN A TWO~DIMENSIONAL ABELIAN SANDPILE 1731

for closed

boundary

conditions

iJ'j(0)

=

~ ~

= 0.11338

(4)

On

moving

inside the

sample,

the

probability iJ'j (I

_{at a} distance

I

_{from the}

boundary

tends to its bulk value

iJ'j(~x~ )

₌

2/7r~ 4/7r~ according

to the law

tPj(ii=tPj(~x~i(i±j+. ), ₍₅₁

where the upper

sign

is related to the open

boundary

conditions and the bottom

sign

to the closed _ones.

The

two-point

correlation function

iJ°jj(r)

at the surface

decays according

_to the law for open

boundary

conditions

iJ°jj(r)

=

4

iJ°)(0) ^~~

^{" ~~ ~~}

~ ~

+,

,

(6) (27

_7r -108 _7r

+64)

_r

for closed

boundary

conditions

£Pjj (r)

=

£P)(o) ~/

~ +

(7)

7r r

1.2 IN THE CASE OF AN INFINITELY LONG STRIP OF WIDTH L. The

leading exponential decay

of the

pair

correlations

iJ°jj(r) along

the central line of the

strip

is obtained.

a)

For both open and closed

boundary

conditions

iP, (r

_; L

~4

=

iJ°)(L/2)

e~ ^{~ "'~~} +

,

(8)

L~

where the finite-size corrections _to the

probability iPj(L/2)

ⁱⁿ the middle of the

strip

_are :

2

iPj(L/2)

=

iJ°j(~x~)(1±)+. ). (9)

4L

Here the upper

sign

is related to the _open

boundary

conditions and bottom

sign

to the closed

ones.

b) For

periodic- boundary

conditions the

pair

correlations take the form

iJ°jj (r

_;

L)

=

iP)(L/2)

^{~ "}

e~~

"'~~

+

,

(lot

L~

~

where

~

~s iJ°,(L/2)

=

£Pj(cc)~l

~

+

(ll)

15(7r 2)L

The above results _can be

compared

^with the

predictions

^of the conformal

field-theory.

Thus, from

(6), (7)

it follows that _xii

= 2, and from

(8)

_one has A

=

2 _7r, which is in agreement ^with

Ill-

For

periodic boundary

conditions

equation (10) yields

A

=

4 _7r, and the bulk exponent

.< =

2 has been obtained

by Majumdar

and Dhar

[12],

which is

again

in

conformity

with the

prediction (1).

Thus _our results confirm the

hypothesis

about the

correspondence

^between the unit

height

in the Abelian

sandpile

model and the energy in the zero-component ^{Potts model. As far} as other

observables _are

concerned,

_e-g-

heights 2, 3, 4,

sizes of avalanches, their duration and

perimeters,

we lack _at present

convincing

evidences of their relation to observables in the Potts model. A short discussion of that

problem

is

given

in the final section.

2. Correlations and Green functions.

The main tool for evaluation of

height probabilities

and correlations between them is _a

mapping

of the _set of

sandpile configurations

onto the set of

spanning

trees. Dhar

[8]

has shown that the number

l"~

of

sandpile configurations

in the SOC state, as well _as the number of

spanning

trees, is

given by

^the

simple expression

~V"~ ₌ ^{det A.}

(12)

The different

height probabilities

_{on a} lattice fit _can be related also to the enumeration of certain types ^of

spanning

trees. In

particular,

the number of

sandpile configurations

with z, =

I at a

given

site I of fit

equals

the number

-Vi''

of

spanning

trees on the _same lattice fit

which have

just

_one leaf attached to and

pointing

in _a fixed direction

[9]. Following

Majumdar

and Dhar

[12],

_{one may} construct _{a new} lattice fit _~'~, such that the

spanning

trees

covering

fit ^~'~ contain the fixed leaf. To this end, one

just

cuts off all the bonds attached _to site I in

fit,

but the _one

containing

the

given

leaf. As _a

result,

_{a new}

toppling

matrix A~' for the

sandpile

on fit ^~'' is obtained. Note that the defect matrix B

= A^~' A has _non-zero

elements

only

for the site and its _nearest

neighbours

but _one. Thus the

probability

of

finding

the value z, = I is

given by

the formula

iJ°j

₌ ^{~~~ ~}

~'~

=

det

(I

₊ GB

), (13)

where I is the unit matrix and G

= A~ is the lattice Green function. For

example,

if I is far away from the boundaries,

B,,

=

3 and

B~~ ⁼ I,

B~~ ⁼ B~, =

I for

j being

the left,

right

and lower

neighbour

of I. In this _case iJ° is

independent

^of the choice of site I inside the bulk of the lattice fit. The

pair

correlation function

Pi (r

) can be defined in _a similar way

by constructing

a new

toppling

matrix A^~'J for the

lattice,

fit ^~'J with defects of the above described type

placed

at a distance _r apart _;

where Bjj

is

_{the compound} defect atrix.

Thus, the problems we have ormulated in

the

introduction can be

reduced

to the

of

the

lattice Green ^unctions for

different

boundary

eterminants

(13)

or

(14)

for the specified

defects.

The

eigenvalues and

eigenfunctions of the

discrete

Laplacian _A on a finite lattice the

rectangularshape nder the consideredboundary

conditions

are _well

known. For

a

_lattice

of

size M

x _L with open boundary ^onditions on the vertical

or

_eriodic

onditions

on

^{the orizontal edges,}

G)~/(nl,

_{ml n2,}

m2)

"

~j ~j

V~~~(nj

)

_Vj~

~(ml)

j~j jj V~~~(n2~

V~~~(~2). (15)

~,~~,

Ap +A~

N° 8 BOUNDARY EFFECTS IN A TWO-DIMENSIONAL ABELIAN SANDPILE 1733

Here

n=1,2,..,L

labels the _row and

m=1,2,...,M

labels the column of site

I

=

(n,

_m

)

_e fit

v~(n

are the

eigenfunctions

and A~ ^{are the}

corresponding eigenvalues

of the one-dimensional discrete

Laplacian

under

boundary

conditions

(T ),

_{T =} I

(open),

2

(closed)

and 3

~periodic).

For the vertical direction _one has the

following explicit expressions

~p ₌ 1, 2,

,

L ; n = 1, 2,

,

L)

:

for _open

boundary

conditions

Al'

₌ 2

Ii

CDS

~l~i vi' ~(n)

-

l~

i

Sin

Ill _('61

for closed

boundary

conditions

~l

^j_

A)~~ =

2

(1

^cos

^I

^;

^v)~J(n

^{) =}

'~

^{~ ~ ~}

~

j~cos (7r~p-1)(n-~ )/Lj, ^{p=2,. ,L;}

L 2

(17)

for

periodic boundary

conditions

Ai~'-

₂

Ii

CDS

~

i~

_;

_vi~>(ni

-

)

~xP

~

t~~ ('81

Passing

to the limit of _an

infinitely long strip,

M

- ~x~, we first redefine the Green function

(15)

_as follows.

ii

Shift the

origin

of the column number _m and the _row number _n to the _center, I,e, for _an odd

integers

M, L _{we set m}

=

m' ₊ ~ ~

n =

n' ₊ ~

where mj, nj labels the columns 2

and _rows from the central _ones

(with

_mj

= 0, _nj

=

0).

ii)

Shift the value of the Green function

by

a constant term to remove the

divergency

^which appears as M

- ~x~, even at finite L when

periodic boundary

conditions are

imposed

_on the horizontal

edges, Evidently,

such _a shift does _not affect the value of the determinants

(13), (14),

It is convenient _{to set} the shift term

equal

to the value of

G)(

_at

coinciding

sites in the

center of the lattice, Then, in the limit M _- ~x~ we obtain

G)~~(Jll, ~~

,

Jl2,

Rli)

"

~

~lffl

~~)i~(Jl~+~)~,~i+~)~,~i+~)~,Rli+~)~

M-oJ

G(Tj

^{L + I M + L + I M +}

~~' 2 2 ^' 2 2

~(T)~~

j

j(Tl(~

j ~~~

(~i

~i j _~

~(Tl

^{~ + l} p(Tl ^{~ + l}

L j

j"

^{P I p 2 2 p ~ p ~}

=

~

^dry

,

(19)

~,=

"

o

A~~J+ ²⁽¹

cos cy

Considering

correlations

along

the central _row of the

strip

it is convenient to represent ^the

resulting

Green function _{as a sum} of two parts an even, C )~~, and an odd,

S)~

^{', with} respect to each of the arguments

n(

and

ii(

=

C[~'(n(, _{n( m( m()}

₊

_{S)~~(n(, n( m( m(), (20)}

In the remainder,

referring

to the

strip

_{geometry, we} shall

drop

the

prime superscript

of the

row and column numbers,

remembering

that

they

_are counted from the center,

Explicitly

_we

have

for open

boundary

conditions

all)(~

~ ~~y)

~(~~

_l

~ ~' ~' L+I ~

j,_~ ^7r

j«

^cos

^{[7r (2 I}

^{+ I}

^{) nil (L}

^{+ I}

^)]

^cos

^{[7r (2 I}

^{+ I}

^{) n~/ (L}

^{+ I}

ⁱⁱ

^{cos ma}

~

o 2 _cos

[7r (2 I

+ I

nil (L

+ I

ii

_{cos cy} ^{~~ ' ~~~~}

S)'~(nj,

_n~

m)

=

j ^{IL 'V2} j

j«

^sin

^{[2 grinj/(L}

^{+ I}

^)]

^sin

^{[2 7rin~/(L}

^{+ I}

ⁱⁱ

^{cos ma}

L ₊ I

~j,

7r ~ 2 cos

[2 7ril

(L + I

ii

_{cos cy} ^{~~ ' ~~~~}

for closed

boundary

conditions

~~~~~" ~~'~~

2

iL ~

~)~ )s ^y~

_{~~ ~}

j ^{IL -'V2}

j"

^cos

^{(2 grinj/L)}

^cos

^{(2 grin~/L)}

^{cos ma}

~ L

~j,

7r ~ 2 _cos

(2 7rilL )

_{cos cy} ^{~~ ' ~~~~}

S)~~(iij,

_n~,

ml

=

i ^{IL ">2}

j«

^sin

^{[7r (2 I}

^I

^nj/L]

^sin

^{[7r (2 I ) n~/L}

^{cos ma}

L

~j~

7r ~ 2 _cos

[7r (2

^I I

)IL

_cos

cy

~~

'

~~~~

for

periodic boundary

conditions

~~~ i ^L j

j«

^sin

^{(2 grinj/L)}

^sin

^{(2 grin~/L )}

^{cos ma}

~~

_{~'~" '~~' ~} 2 L

~j,

gr ~ 2 _cos (2

7rilL)

_cos

cy

~~ ~~~~

In the _case of _a

half-plane

geometry, one may ^start with

expression (19)

and take the limit L

- ~x~, thus

obtaining

_{up to}

unimportant

constant the Green function for _a

half-plane

with open

boundary,

~, j

j« j«

^{sin nj}

^fl

^{sin n~ fl cos ma}

G

'(nj,

_{n~ ;}

m)

= j dry

dfl

,

(27)

w ~ ~ 2 _cos fl cos _cy

and closed

boundary

~~~

i

In j«

^cos

^{(nj 1/2) fl}

^cos

^{(n~ 1/2)}

^{fl cos ma}

~ ~'~~' '~~ ~

w2

~ ~ 2 _cos fl cos a

~~

~~

~~~~

N° 8 BOUNDARY EFFECTS IN A TWO-DIMENSIONAL ABELIAN SANDPILE 1735

3. The

strip

geometry.

We consider the correlations between _two unit

heights,

_{one at} the _center of the

strip,

ni "

0,

_m

= 0, and the other at a distance _r apart, at site n~ =

0,

_m~

= r, If _we label the _rows and columns of the 8 _x 8 matrix B

(r),

associated with the

corresponding

defects, in the order (- 1,

0), (0, Ii, (0, 0), (0, Ii, (-

I,

r), (0,

_r

1), (0, r), (0,

r + I

),

it takes the block-

diagonal

form

Bjj

=

~~ (29)

where

Bj

is the 4 _x 4 matrix

0

1

~~

~~~~

0 0

The restriction

©~(r)

of the lattice Green function

(19) (for simplicity

of notation, in the

remainder of this section _we omit the

superscript (T) indicating

the

specific boundary

conditions)

to the

corresponding

set of defect sites is _an 8 _x 8 matrix with the block _structure

/j

_~~~

^{AL(0) AL(r)}

~

AL(-

?')

L(0)~

'

(3')

where,

using

the

representation (20), A~(r)

_may ^{be written} as the 4 _x 4 matrix

C~(I,

_; r) +

S~(I,

I

r) C~(1,

0

r I

) C~(1,

⁰ ;

r) C~(1,

⁰ ; r +

ll

~~~'~~

~~[

~j~~ ^~~ ~~~~[~j~~-1) ~~~~[~j~~

^~~

~~~[~j~~~~~

C~(0,1;r-I) C~(0,0;r-2) C~(0,0;r-I) C~(0,0;r)

(32)

and

A~(0)

is

given by (32)

at r =

0.

According

to

(14),

the

pair

correlation function

Pi j(r

L) is

iJ'jj

(r _;

Li

= det 11 +

©~(r) _Bj (rii iJ'I(L/2), (33)

where

iJ°)(L/2)

=

lim det

[1

₊

©~(r) _{Bjj (r)}

,

(34)

- a~

is the square of the unit

height probability

at the central line of the

strip,

Thus the

problem

reduces _to the evaluation of the determinant

~~~ ~ + GL

(?')

B

(r ii

= det

~ _~

~L ~°

l ~

' AL

(?'I

B

~L(- ")

B I ₊

AL(-

_r

Bj ^~~~)

Using

the

explicit

form of

Bj

and the

symmetry properties

of the _even and odd

components

of the Green function

(20),

after obvious

transformations,

_we obtain the

expression

det

[1+ ©~(r)Bjj(r)]

=

6~ S~(q,

1;

0)j~ ^S)(I,

^I

r))

^det

f

~~~l)

,

(36)

R

(r) Q

where

Q

^and

R(r)

_are the 2 _x 2 matrices

(R~

is the transpose ^of

RI,

C~ (0,

0

C~(0,

⁰ ;

C~(0,

_;

~ II

C~(0,

0 _;

1) C~ (0, 1)

C

~(0,

0

)

^{' ~~~~}

C~(0,0;r)-C~(0,1;r)

C~(0,0;r+I)-C~(0,1;r+I)j

~ ~~

C~(0,

0 _r I

C~(0,

_r I

C~(0,

0 _r

C~(0,

0) ~~~~

The

I-dependent

terms in

equations (36), (38)

_are

readily

evaluated _{as r- ~x~} at fixed L » I _; their

leading asymptotic

form

depends

_on the

boundary

conditions

I)

For open

boundary

conditions

S~(I,

I _r

=

~

) _exp(-

_{2 ml-IL )}

₍₃₉₎

L

C~(0,

0

r) C~(0, r)

=

~

_exp

_(-

_grr/L

;

(40)

2 L

2)

For closed

boundary

conditions

S~(I,

I

r)

=

~ _{exp(- grilL) (41)}

L

C~(0,

0

r) C~(0,

_r

=

~ _exp(-

_{2 7rr/L}

₍₄₂₎

L

3)

For

periodic boundary

conditions

S~(I,

_{I ; r}

=

~

)

exp

(-

2 7rr/L )

(43)

L

C~(0,

0

r) C~(0,

_r

=

~ _exp(-

_{2 7rilL}

_{) (44)}

L

By

substitution of

(40)-(44)

ⁱⁿ

expression (36),

see also

(33),

_one obtains that the

exponential decay

of correlations is

governed by

the correlation

length f

₌ L/2 _7r in the _cases

of open and closed

boundary conditions,

and f

=

L/4 _7r in the _case of

periodic boundary

conditions. The finite-size corrections _to the unit

height probability iJ°j(L/2),

see

(34),

iJ°j(L/2)

=

[1+ C~(0,

0 _;

2)][1+

2

C~(0,

1;

Iii( S~(I,

1;

0)) ⁽⁴⁵⁾

can be obtained

by using

^standard

techniques

^based on the Poisson summation formula, see

e-g-

[16].

All the relevant terms can be

expressed through

four

L-dependent integrals F~(L),

k

= 1,

,

4,

i ii

i i ^{ii ii}

ii ^il+~iii~i~ _iii

N° 8 BOUNDARY EFFECTS IN A TWO-DIMENSIONAL ABELIAN SANDPILE 1737

where,

&L(~b =

I(sin

_{~b +}

~jL

I

j-

^'

(soy

Thus _we obtain _:

Ii

For open

boundary

conditions

C~(0,0;2)=

-1+ ~

-4sj(2L+2)-4 s~(2L+2) (51)

C~(0,

_;

1)

= + 2

s~(2

L ₊

2) (52)

S~(I,1;0)= -2s~(2L+2)-2s~(2L+2), (53)

2)

For closed

boundary

conditions

C~(0,

0

2)

= +

~

4

e~(2 L)

₊ 4

s~(2

L

) (54)

C~(0,

_; 1)

=

2

s~(2

L

(55)

S~(I,

1;

0)

= + 2

Fj(2L)

+ 2

s~(2 L). (56)

3)

For

periodic boundary

conditions

C~(0,

0 _; 2)

= +

~

4

s~(2

L) + 4

E~(2

L

(57)

C~(0,

_;

1)

= 2

s~(2

L

(58)

S~(I,

1;

0)

=

2

s~(2L)

2

s~(2L). (59)

The final results for the

strip

_{geometry are} summarized in

equations (8)-(11),

see the introduction,

4. The

half-plane geometry.

With the aid of

expressions (27)-(28)

for the Green

functions,

_{one can use} the determinant formula

(13)

to obtain the unit

height probability

at a site which

belongs

_to the lattice

boundary

3fif. In this _case, the modification of the lattice consists in

cutting

off the _two bonds which _are attached _to I and

belong

to the

boundary.

If _we label the defect sites in the order (- 1, ⁰

), (0,

0

), (1,

0

),

the matrix associated with the defect is the 3 _x 3 matrix of the form _: for _{an open}

boundary

0

B)'J

=

3

,

(60)

0

for _a closed

boundary

0 B(~~ =

l 2

(61)

0

A direct evaluation of the

corresponding

determinants leads to

expressions (3)

and

(4), given

in the introduction.

Of _{course, one} could _{construct a} different defect matrix which leads _to the _same

results,

for

example, by preserving

_one of the

boundary

bonds and

cutting off,

instead of

it,

the bond which _starts from I and

points

inside the lattice.

To obtain _{a more} detailed information about the

spatial

structure of the

sandpile,

_we have studied also the

asymptotic

behaviour of the unit

height probability iJ°j (I)

_at

_large

_distances

from the

boundary, ^I

» I. In this _case, the defect matrix is the _{same as} in the

bulk,

_see

[12].

The

asymptotic

behavior of the

boundary

Green function at

large separations,

for both open

and closed

boundary conditions,

is _most

easily

obtained with the aid of the method of

reflections. Then, one needs _to know

only

the

asymptotic expansion

for the bulk Green

function at

large separations (r

»

1)

_:

G~~j~(0,

⁰ ;

r)

= ln _r

$

^{~ ~~ ~}

+ +

~~

~

+

(62)

2 _" 2 _" 4 _" 24

grr~

480 _grr

where _y is the Euler constant, and

simple

calculations

give

the result

(5).

By using

the discrete

Laplace equation

_{to generate a sequence} of recurrent

relations,

_one

may obtain the

asymptotic expansion

for

G~~j~(0,n;r),

where _n

=1,2,... Next, by

subtracting (in

the _case of _an open

boundary)

_or

adding (in

the _case of _a closed

boundary)

the Green function for _a fictitious _source,

placed symmetrically

with respect to the

boundary,

_one

can

easily

find the

asymptotic expansion

for the Green function which

corresponds

_to the considered

boundary problem,

Thus for _open

boundary

conditions _we obtain

G~'~(0,0;r)=(-j+.

,

(63)

grr 2grr

and for closed _ones

G~~~(0,0;r)=- lnr-1-~~~~-

~-

~~

~+. ⁽⁶⁴⁾

" " 2

gr 6 _grr 240 _grr

The defect matrix used in the calculation of the unit

height

correlations at the

boundary,

in the

representation

when the defect sites _are labeled in the order:

(-1,0), (0,0), (1,

0

), (r

1, 0 ),

(r, 0), (r

₊ 1, 0

),

has the

block-diagonal

forrn

Bii

~

~~

,

(65)

where the 3 _x 3 matrix

Bj

is

given by

one of the

equations (60), (61).

The restriction of the

boundary

Green function _to the set of defect sites _can be written in the block form

A~ (0 )

AL

(ri

©L(ri (661

~

~~~- ii AL(01

N° 8 BOUNDARY EFFECTS IN A TWO-DIMENSIONAL ABELIAN SANDPILE 1739

Finally

the evaluation of the determinants

(14)

leads _{us to} the results

(6), (7) presented

in the introduction.

5. Conclusion.

In summary, we have calculated the

leading asymptotic

form of the correlation functions for the unit

heights, iJ°j (r

), in the

strip

and

half-plane geometries

of the Abelian

sandpile

model,

under three different

boundary

conditions. The results confirrn the

suggestion

that the unit

height

behaves like the local energy operator ^{in the} q-component ^{Potts model} at q =

0.

Unfortunately,

the above conclusion cannot be

directly

extended _to the _case of

heights

z, = 2,

3, 4, although correspondences

between these

heights

and

spanning

trees still exist. In

[17]

it has been shown that _z,

=

2

corresponds

to such _trees

(in

addition _to the trees

yielding

z, = I

),

in which _none of the

paths, starting

from three nearest

neighbours

of site I and

ending

at the open

boundary,

_pass

through

I. In the _case z, =

3 this property ^{must have the}

paths starting

from two of the _nearest

neighbours

of I, and if z, = 4 from _one

neighbouring

site,

Thus,

the

heights

_z,

=

2, 3, 4, being

local

sandpile

observables,

happen

to be connected with nonlocal

properties

of the trees. For the enumeration of

configurations

with these

heights

_one

has to consider clusters of subtrees with

increasing

size. It has been established

[12]

that such

cluster

expansions

are

slowly

convergent, ^{which makes difficult the evaluation of the}

correlation functions

P~~(r)

for _z

= 2, 3, 4. In

principle,

the above mentioned features may lead to _a different from

r~~

_power law of

decay.

Finding

a

correspondence

between observables in the Potts model and characteristics of the avalanches is not an easier task

either,

because of the nonlocal _nature of the avalanches. The solution of these

problems

needs further

analytical

and numerical studies.

6.

Acknowledgements.

This work _was

supported

in part

by

the

Bulgarian

National Foundation for Scientific Research, Grant No. MM-74/91.

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