5
Перколяция в конечной полосе для гиббсовских решеточных моделей
( )
( , )
Ȗ
Ȗ
1
( 1)
( , , ) exp{
(
)},
Ȗ
l
j
j
j
Z S h p
k
k
J
/ *
*
*/
*
*
*
¦
¦
ɝɞɟ
/
=
S
u
h
, ɜɬɨɪɚɹ ɫɭɦɦɚ ɛɟɪɟɬɫɹ ɩɨ ɜɫɟɦ
J
, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɦ
ɭɫɥɨɜɢɹɦ, ɚɧɚɥɨɝɢɱɧɵɦ ɭɫɥɨɜɢɹɦ 1–2 § 1.4 ɜ [6], ɬɚɤɢɦ, ɱɬɨ ɥɢɛɨ
1
,
B
*
ɥɢɛɨ
B
1
ɥɟɠɢɬ ɧɢɠɟ
*
[6–7].
ɗɬɨ ɩɨɥɧɵɣ ɚɧɚɥɨɝ ɤɥɚɫɬɟɪɧɨɝɨ ɪɚɡɥɨɠɟɧɢɹ ɥɨɝɚɪɢɮɦɚ
ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɫɭɦɦɵ ɜ [7] ɞɥɹ ɧɚɲɟɝɨ ɚɧɫɚɦɛɥɹ ɦɧɨɠɟɫɬɜ. ɂɡ
ɥɟɦɦɵ 1
(Ȗ)
( , )
2
2 2
Ȗ
Ȗ
( 1)
( , , ) exp{
}.
Ȗ
l
H S h p
k
k k
/ *
*
*
/
*/ */
*
*
¦ ¦ ¦
ɋɭɦɦɢɪɨɜɚɧɢɟ ɜ ɷɤɫɩɨɧɟɧɬɟ ɢɞɟɬ ɩɨ ɜɫɟɦ ɧɚɛɨɪɚɦ
*
ɢ (
*
,
J
), ɜ
ɤɚɠɞɨɦ ɢɡ ɤɨɬɨɪɵɯ ɟɫɬɶ ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɨɞɢɧ ɤɨɧɬɭɪ ɩɪɨɬɟɤɚɧɢɹ
*
'
R
. Ɉɬɫɸɞɚ ɥɟɝɤɨ ɫɥɟɞɭɟɬ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 1.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 2
. ɉɪɨɜɟɞɟɦ ɟɞɢɧɢɱɧɵɣ ɝɢɩɟɪɤɭɛ
ɱɟɪɟɡ ɫɟɪɟɞɢɧɭ ɨɬɪɟɡɤɚ, ɫɨɟɞɢɧɹɸɳɟɝɨ ɫɨɫɟɞɧɢɟ ɫɩɢɧɵ
ı ,ı ,
t
t
c
1, ,
,
t t
t t
c
c/
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɷɬɨɦɭ ɨɬɪɟɡɤɭ, ɟɫɥɢ ɫɩɢɧɵ
ɢɦɟɸɬ ɪɚɡɧɵɟ ɡɧɚɤɢ. Ɇɧɨɠɟɫɬɜɨ ɩɨɥɭɱɟɧɧɵɯ ɝɢɩɟɪɤɭɛɨɜ ɦɨɠɧɨ
ɨɞɧɨɡɧɚɱɧɨ ɪɚɡɛɢɬɶ ɜ ɨɛɴɟɞɢɧɟɧɢɟ ɫɜɹɡɧɵɯ ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ
(ɩɟɪɟɫɟɱɟɧɢɟ ɩɨ ɤɭɛɚɦ ɪɚɡɦɟɪɧɨɫɬɢ ɦɟɧɶɲɟ
v
ɧɟ ɛɟɪɟɦ ɜɨ ɜɧɢɦɚɧɢɟ)
ɤɨɦɩɨɧɟɧɬ, ɹɜɥɹɸɳɢɯɫɹ ɫɜɹɡɧɵɦɢ ɡɚɦɤɧɭɬɵɦɢ ɛɟɡ ɫɚɦɨɩɟɪɟɫɟɱɟɧɢɣ
ɝɢɩɟɪɩɨɜɟɪɯɧɨɫɬɹɦɢ, — ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɢɯ ɞɭɚɥɶɧɵɦɢ ɤɨɧɬɭɪɚɦɢ, ɢ
ɪɚɡɞɟɥɹɸɳɢɯ «ɦɨɪɹ» ɩɥɸɫɨɜ ɢ «ɨɫɬɪɨɜɚ» ɦɢɧɭɫɨɜ ɬɚɤ, ɱɬɨ
«ɨɫɬɪɨɜɚ» ɦɢɧɭɫɨɜ ɨɛɪɚɡɭɸɬ ɨɞɧɨɫɜɹɡɧɵɟ ɦɧɨɠɟɫɬɜɚ. Ⱦɭɚɥɶɧɵɣ
ɤɨɧɬɭɪ, ɫɨɟɞɢɧɹɸɳɢɣ ɨɛɚ ɨɫɧɨɜɚɧɢɹ
/
S, h
, ɧɚɡɨɜɟɦ ɞɭɚɥɶɧɵɦ
ɤɨɧɬɭɪɨɦ ɩɪɨɬɟɤɚɧɢɹ, ɢ ɦɧɨɠɟɫɬɜɨ ɬɚɤɢɯ ɤɨɧɬɭɪɨɜ ɨɛɨɡɧɚɱɢɦ, ɤɚɤ ɢ
ɩɪɟɠɞɟ, ɱɟɪɟɡ
R
. Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɜ ɬɟɪɦɢɧɚɯ ɞɭɚɥɶɧɨɣ ɤɨɧɬɭɪɧɨɣ
ɦɨɞɟɥɢ
H
(
S
,
h
,
E
) ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɚɦɢ (3) ɢ (4), ɝɞɟ ɫɭɦɦɢɪɨɜɚɧɢɟ
ɢɞɟɬ ɩɨ ɜɫɟɦ ɞɨɩɭɫɬɢɦɵɦ ɞɭɚɥɶɧɵɦ ɤɨɧɮɢɝɭɪɚɰɢɹɦ ɫ ɜɟɫɚɦɢ
2ȕ
(2)
,
k e
*
*
(1)
(2)
,
k k
*
*
ɟɫɥɢ
,
R
*
ɢ
(1)
0,
k
*
ɟɫɥɢ
,
R
*
*
— ɱɢɫ-
ɥɨ ɟɞɢɧɢɱɧɵɯ ɝɢɩɟɪɤɭɛɨɜ ɜ
*
. Ⱦɚɥɶɧɟɣɲɢɟ ɨɰɟɧɤɢ ɩɪɨɜɨɞɹɬɫɹ ɬɚɤ
ɠɟ, ɤɚɤ ɢ ɞɥɹ ɧɟɡɚɜɢɫɢɦɨɝɨ ɩɨɥɹ. ɋɨɜɟɪɲɟɧɧɨ ɚɧɚɥɨɝɢɱɧɨ ɪɟɲɚɸɬɫɹ
ɜ ɤɨɧɟɱɧɨɣ ɩɨɥɨɫɟ ɩɪɢ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ
p
ɢ
E
*
:
ɚ) ɡɚɞɚɱɚ ɧɚɯɨɠɞɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɩɪɨɬɟɤɚɧɢɹ, ɟɫɥɢ ɤɨɧɬɭɪ
ɩɪɨɬɟɤɚɧɢɹ ɢɦɟɟɬ ɜɨ ɜɫɹɤɨɦ ɫɟɱɟɧɢɢ, ɩɚɪɚɥɥɟɥɶɧɨɦ ɨɫɧɨɜɚɧɢɸ,
ɨɞɧɨɫɜɹɡɧɭɸ ɨɛɥɚɫɬɶ ɩɥɨɳɚɞɶɸ ɧɟ ɦɟɧɟɟ
r
.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɧɚɩɪɢɦɟɪ, ɞɥɹ
v
= 1,
/
=
L
u
h
:
