3
Перколяция в конечной полосе для гиббсовских решеточных моделей
Ɍɟɨɪɟɦɚ 2
. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɯ
E
>
E
0
* 2 2 2
* 2( 1)
*
( , ,ȕ) exp{ (ȕ )
[1 (ȕ )
ij( , ,ȕ )]},
h h
H S h
S
S h
Q
Q
(2)
*
1
( , , )
,
h
S h
Q
Q
M E
ɝɞɟ
E
*
=
e
–2
E
,
M
(
S
,
h
,
E
*
) — ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɩɪɢ
*
*
0
ȕ ȕ ;
Q
v
— ɤɨɧɫɬɚɧɬɚ, ɡɚɜɢɫɹɳɚɹ ɥɢɲɶ ɨɬ ɪɚɡɦɟɪɧɨɫɬɢ
v
.
ȼ ɮɨɪɦɭɥɚɯ (1–2) ɥɟɝɤɨ ɜɵɱɢɫɥɢɬɶ ɫɥɟɞɭɸɳɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
ɩɪɢ
p ˾
h
+1
, (
E
*
)2
(
v
–1)
h
+2
v
ɢ ɬ. ɞ., ɧɨ ɨɧɢ ɭɠɟ ɛɭɞɭɬ ɡɚɜɢɫɟɬɶ ɨɬ ɜɢɞɚ
ɨɫɧɨɜɚɧɢɹ
S
.
ɉɭɫɬɶ, ɧɚɩɪɢɦɟɪ,
L h
/ u
,
1
L
]
, ɬɨɝɞɚ
1
2
ln ( , , )
[4( 1)( 1) (3 2 )]
(
)
h
h
h
H S h p Lp
L h
Lh h p LO p
ɜ ɫɥɭɱɚɟ ɧɟɡɚɜɢɫɢɦɨɝɨ ɩɨɥɹ ɢ
* 2 2
* 2 4
* 2 6
ln ( , ,ȕ)
(ȕ )
4( 1)( 1)(ȕ )
((ȕ ) )
h
h
h
H S h
L
L h
LO
ɜ ɫɥɭɱɚɟ ɦɨɞɟɥɢ ɂɡɢɧɝɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ
v
= 3 (ɞɥɹ ɩɪɨɫɬɨɬɵ) ɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɨɛɴɟɦɨɜ
/
n
=
L
n
u
M
n
u
h
ɬɚɤɭɸ, ɱɬɨ ɩɪɢ
n
ĺ
f
/
n
ɡɚɩɨɥɧɹɟɬ ɛɟɫ-
ɤɨɧɟɱɧɭɸ ɩɨɥɨɫɤɭ
2
h
u
]
. Ɉɩɪɟɞɟɥɢɦ ɜ ɤɚɠɞɨɦ ɨɛɴɟɦɟ ɫɥɭɱɚɣɧɵɟ
ɩɨɥɹ — ɧɟɡɚɜɢɫɢɦɵɟ ɫ ɩɚɪɚɦɟɬɪɨɦ
p
n
ɢ ɝɢɛɛɫɨɜɫɤɢɟ (ɢɡɢɧɝɨɜɫɤɢɟ) ɫ
ɩɚɪɚɦɟɬɪɨɦ
E
n
— ɢ ɱɟɪɟɡ
]
n
ɨɛɨɡɧɚɱɢɦ ɱɢɫɥɨ ɤɨɧɬɭɪɨɜ ɩɪɨɬɟɤɚɧɢɹ ɜ
ɤɨɧɮɢɝɭɪɚɰɢɢ
/
n
. ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ, ɱɬɨ ɩɪɟɞɟɥɶɧɵɣ ɩɟɪɟɯɨɞ
ɬɚɤɨɣ, ɱɬɨ
Ȝ
h
n n n
M L p
o
(ɜ ɫɥɭɱɚɟ ɧɟɡɚɜɢɫɢɦɨɝɨ ɩɨɥɹ) ɢ
* 4 2
(ȕ )
Ȝ
h
n n
n
M L
o
(ɜ ɫɥɭɱɚɟ ɝɢɛɛɫɨɜɫɤɨɝɨ ɩɨɥɹ).
Ɍɨɝɞɚ ɜɟɪɧɵ ɫɥɟɞɭɸɳɢɟ ɬɟɨɪɟɦɵ.
Ɍɟɨɪɟɦɚ 3
. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ
P
n
(
]
n
= l) ɫɯɨɞɢɬɫɹ ɤ
ɩɭɚɫɫɨɧɨɜɫɤɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɸ:
Ȝ
Ȝ
(ȗ )
.
!
l
n n
e
P l
l
o
Ɍɟɨɪɟɦɚ 4
. ɉɭɫɬɶ
O
— ɩɪɨɢɡɜɨɥɶɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ. Ɍɨɝɞɚ
ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɦɨɞɟɥɟɣ ɫɭɳɟɫɬɜɭɟɬ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ
p
<
p
0
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
E
*
<
E
0
*
)
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɨɛɴɟɦɨɜ
,
n
n n
S h
Q
/ u
]
,
n n
S
of
of
,
n n
h
of
of
ɬɚɤɚɹ, ɱɬɨ
