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THE
PROBLEM
RANDOM
OF
FIELD
PHASE
R.
L.
OF
UNIQUENESS
AND
THE
OF
A GIBBSIAN
PROBLEM
TRANSITIONS
Dobrushin
1. In a p r e v i o u s p a p e r b y the a u t h o r [1] the f o l l o w i n g c o n c e p t s w e r e i n t r o d u c e d . It w a s a s s u m e d
t h a t the p o t e n t i a l U(t), t E T V, w a s s p e c i f i e d , w h e r e T u is a P - d i m e n s i o n a l i n t e g e r l a t t i c e s u c h t h a t a) f o r a
eel"tain d < ~ we h a v e
Y, IU(t)l<
~°
(1.1)
t:!ti~d
and b) U(t) = U ( - t ) , t 6 T v, the c h e m i c a l p o t e n t i a l / 2 , s a t i s f i e s the c o n d i t i o n s - ~ </2 < ~, and the c o n s t a n t / 3 ,
s a t i s f i e s the c o n d i t i o n 0 < /3 < ~o, (this c o n s t a n t is i n v e r s e l y p r o p o r t i o n a l to the t e m p e r a t u r e ) . The r a n d o m
f i e l d ~(t), t E T v, which h a s the v a l u e s in the s p a c e X c o n s i s t i n g of two p o i n t s 0 and 1, is s p e c i f i e d b y a s e t
of f i n i t e l y - d i m e n s i o n e d d i s t r i b u t i o n s P = ~PV(Xl . . . . .
x IvI ), v c TV}, w h e r e V is a f i n i t e s e t of I V I e l e m e n t s and xi E X. T h e r a n d o m f i e l d and i t s d i s t r i b u t i o n a r e c a l l e d " G i b b s i a n " i f the c o n d i t i o n a l p r o b a b i l i t y
s a t i s f i e s the c o n d i t i o n
P {~ (t,) = x, . . . . .
~ (t]v0 = x l v , / ~ (t) =x(t), t E T ~ \ V }
with p r o b a b i l i t y 1 f o r a l l f i n i t e V = {t 1. . . . .
with v a l u e s in X; h e r e
qv (x~ . . . . .
tIvI} c
TV, x l E X . . . . .
~xp { - ~Uv (x, . . . . .
xlvt/x (t)) =
= qv (x, . . . . .
.~
xlvl/x (t))
(1.2)
x l v I EX and f u n c t i o n s x(t), t E T V \ V ,
~lvl,,.~ (t))}
(1.3)
~lvl/~ (t))}
~xp { - 13uv(.,-,. . . . .
x~EX,...,XlVi6x
and
IVl
I Iyl
Uv(x, . . . . . Xlvl/x(t)) - - - - I ~ ~ x, L ~.. ~,
i=1
~
ivl
.v,xy (&--ti) 4- ~,
i=~ ]=l, ]~ei
~, X,X (I)U (t,---t).
(1.4)
i~1 tET~'~,V
A GibbsiandistributioninavesselVwith
t h e b o u n d a r y c o n d i t i o n s x(t), t E T V ~ V , is d e f i n e d a s a r a n d o m
f i e l d ~(t) s u c h that
P {~ (tl) : xt . . . . . ~ (/Ivj) =: xlvl} = qv (xl . . . . . Xlvl/x (t)),
P {~ (t) = x
(t)) = 1, t ~ T v \ V .
(1.5)
In [1] c o n c e p t s w e r e d e v e l o p e d d e m o n s t r a t i n g the f a c t that G i b b s i a n d i s t r i b u t i o n s s p e c i f y m e t h o d s of d e s c r i b i n g o h y s i c a l s y s t e m s in an " i n f i n i t e " v e s s e l . It w a s a l s o s h o w n t h a t the c a s e when the G i b b s i a n d i s t r i b u t i o n with the s p e c i f i e d p a r a m e t e r s (/2, /3, U(')) is unique c o r r e s p o n d s to the c a s e w h e n t h e r e is no
s e p a r a t i o n of p h a s e s .
T h e p u r p o s e of t h i s p a p e r is to i n d i c a t e the e x p l i c i t c o n d i t i o n s f o r w h i c h it is p o s s i b l e to e s t a b l i s h
t h e u n i q u e n e s s o r , c o n v e r s e l y , t h e n o n u n i q u e n e s s of a G i b b s i a n d i s t r i b u t i o n w i t h s p e c i f i e d p a r a m e t e r s ;
w e a l s o u s e e x a m p l e s o f n o n u n i q u e n e s s to d i s c u s s t h e c o n s e q u e n c e s t h a t d e r i v e f r o m it with r e s p e c t to
p h a s e t r a n s i t i o n s in the s y s t e m .
2.
F i r s t we i n d i c a t e t h e c o n d i t i o n s g o v e r n i n g u n i q u e n e s s of the G i b b s i a n d i s t r i b u t i o n .
T H E O R E M 1. A s s u m e
I n s t i t u t e o f I n f o r m a t i o n - T r a n s m i s s i o n P r o b l e m s , A c a d e m y o f S c i e n c e s of t h e USSR. T r a n s l a t e d
f r o m F u n k t s i o n a l ' n y i A n a l i z i Ego P r i l o z h e n i y a , Vol, 2, No. 4, p p . 4 4 - 5 7 , O c t o b e r - D e c e m b e r , 1968.
O r i g i n a l a r t i c l e s u b m i t t e d F e b r u a r y 23, 1968.
302
l
-2
\'
"~!q{o}(x/x(l))--q{o}(x/'((t))I~t,
sup
~ ' s - - - ~ x ' t ) ' ~x (tJ: x .t i = ~ ( t ~ ,
s(ZTv.
(2.1)
t~-s x~'.¥
w h e r e {0} is a s e t of one point 0 and the u p p e r b o u n d a r y is taken o v e r all p a i r s of the functions x(t), ~(t),
t E T u N {0}, such that x(t) = ~(t), t ~ s. Then t h e r e exists only one Gibbsian d i s t r i b u t i o n . The p r o p e r t i e s
of d i m e n s i o n a l r e g u l a r i t y f r o m within and f r o m without a r e s a t i s f i e d f o r it (see [1]). If the potential U(t)
is finite (i.e., U(t) = 0, Itl > C), then the functions ~oV(') and ¢~¢(.) in (5.3) and (5.4) taken f r o m [1] m u s t be
a s s u m e d equal:
(~. (d) - e-'~vJ,
tp-¢ (d) := e - ~ a ,
~v > 0 ,
~- >0.
(2.2)
If the potential U(t) <¢o f o r all t E T v, then condition (2.1) is c e r t a i n l y s a t i s f i e d f o r the v a l u e s of (Is, fl) s u c h
that 1/1 ~ >-- is0 o r B -< fl0, w h e r e is0 < ¢¢, fl0 > 0 a r e c e r t a i n c o n s t a n t s . F o r an a r b i t r a r y potential U(t) C o n d i tion (2.1) is s a t i s f i e d f o r v a l u e s (is, fl) such that Is -< Is0(/3), w h e r e the continuous function ~0(fl) has a finite
l i m i t for fl -* oo and w h e r e Is0(fl) ~ c//3 f o r fl --* 0, w h e r e the c o n s t a n t c > 0.
P r o o f . The b a s i c p o s t u l a t e of the t h e o r e m f o r m u l a t e d above is a p a r t i c u l a r c a s e of a m o r e g e n e r a l
t h e o r e m (in which it is not a s s u m e d that the conditional d i s t r i b u t i o n s a r e Gibbsian), which was p r o v e d in
detail in [2]. H e r e we b r i e f l y indicate only the b a s i c f o r m u l a t i o n u s e d in its proof; we a s s u m e f o r s i m p l i c i t y that the potential is finite. E a c h point t E T v is j u x t a p o s e d with an o p e r a t o r Qt that o p e r a t e s in the
d i s t r i b u t i o n s p a c e of the r a n d o m fields and c o n v e r t s the d i s t r i b u t i o n P = {PV('), V C. T v} into the d i s t r i b u tion P = {PV('), V C T v. We will define that o p e r a t o r on the b a s i s of the p r o p o s i t i o n that
"Pv (x~ . . . . .
xlvl) --
Pv(xl . . . . . xlvl), t f V ,
P v \ {t} (Xl . . . . . xlvl-~) q{ t} (xlvl/x~ . . . . .
V := {it . . . . . t~vl-t, /}, d ( R V ~ V ,
xlvl-~),
{t})>C,
(2.3)
w h e r e q {t} (x I V I / x l . . . . .
xl V] - 1) is the value of q{t} (x [V I/x(t)), that is g e n e r a l a c c o r d i n g to the a s s u m p tion of f i n i t e n e s s for all functions x(t) with x(ti) = xi, i = 1 . . . . .
IVI - 1 . F o r the r e m a i n i n g V the value of
PV(') is d e t e r m i n e d f r o m the m a t c h i n g condition f o r the finitely d i m e n s i o n e d d i s t r i b u t i o n s (see (2.3) in [1]).
It is obvious that the G i b b s i a n d i s t r i b u t i o n s a r e i n v a r i a n t r e l a t i v e to the o p e r a t o r s Qt. A s s u m e
Q : H Qt.
(2.4)
tF.Tv
(The s e q u e n c e in which the n o n c o m m u t a t i v e o p e r a t o r s Qt a r e multiplied is not essential.) We a r e able to
p r o v e that f o r condition (2.1) the o p e r a t o r Q is c o m p r e s s i v e (in a c e r t a i n g e n e r a l i z e d s e n s e ) , whence t h e
s t a t e m e n t of the t h e o r e m follows.
H e r e we need m e r e l y e s t a b l i s h the r a n g e of v a l u e s of(Is,/3) f o r w h i c h condition (2.1) is s a t i s f i e d .
F o r this p u r p o s e we note that in a c c o r d a n c e with (1.3) we have the following r e s u l t f o r x(t) = x~t), t ~ s,
a n d U ( t ) < ~ , t 6 T v,
t~o
~.~
a
t~o
(2.5)
( i + exp {-- [3 [ -- I-t+ t~'~o] U (t) I 1})~
Since f r o m (1.1) it follows that
[ 1- - exp ( - - ~V (s)} l < oc,
(2.6)
s:~e0
and the s u m of that s e r i e s tends to z e r o f o r / 3 ~ O, it follows f r o m (2.5) that condition (2.1) is s a t i s f i e d for
/3 -<~0. Since Ix + y J - - ~xl [yl, if I x ] - > 2 , y -> 2 and II-eXl -< glxl , w h e r e g < co, i f e x -- 2, it follows that
303
~, I l--expf--PU(s)} I< g7
-F-
~,
~
IU(s)l
exp{-(qu(,,)l}<~
s~o
exp{[email protected])}~ g , ~
exp{--~l:](s)l}>~2
IU(s)l + exp li~ ~ [U(s)[}.
sw-o
~
(2.7)
s~-o
F r o m (2.7) and (2.5) it f o l l o w s that f o r f l ~ oo and s u f f i c i e n t l y l a r g e IpI the s u m with r e s p e c t to s of the r i g h t
s i d e s in (2.5) t e n d s to z e r o (for l a r g e p o s i t i v e it due to the f a c t t h a t the n u m e r a t o r of the f r a c t i o n t e n d s to
~o m o r e s l o w l y t h a n its d e n o m i n a t o r , and f o r l a r g e n e g a t i v e / ~ due to the fact that t h e n u m e r a t o r t e n d s to 0
a n d the d e n o m i n a t o r t e n d s to 1). T h e r e f o r e , C o n d i t i o n (2.1) is s a t i s f i e d f o r litl ->itn.
When U(t) c a n a c q u i r e the v a l u e s oo, I n e q u a l i t y (2.5) and the s u b s e q u e n t e s t i m a t e s a r e a p p l i c a b l e if
x(t) = x(t) = 0 w h e r e v e r U(t) = ~ . If x(t) = x(t) = 1 f o r U(t)
~o, then the l e f t s i d e in (2.5) is s i m p l y e q u a l to
z e r o . T h e r e s t i l l r e m a i n s the c a s e when U(s) = ~o, x(s) = l , ~ ( s ) = 0 and x(t) = ~ ( t ) = 0 f o r U(t) = ~o, t ~ s .
For that case
"l
-7-
x~x
[ [
(#z(°)l:=
t-l-exp --~ - - ,
t~o
t~o
(2.s)
J)
and t h i s e x p r e s s i o n t e n d s to 0 f o r it --- -oo u n i f o r m l y with r e s p e c t to x(t), ~(t) a n d a l l s u f f i c i e n t l y l a r g e ~.
F i n a l l y , f o r fl ~ 0, #fl = c o n s t the e x p r e s s i o n in (2.8) t e n d s to eflit/(1 +efl/~), w h e n c e it f o l l o w s thatPo(~)~c/fl.
It c a n b e s t a t e d c o n v e n i e n t l y t h a t C o n d i t i o n (2.1) m e a n s t h a t the c o n d i t i o n a l G i b b s i a n p r o b a b i l i t i e s
d e p e n d w e a k l y on t h e c o n d i t i o n s . T h e s a t i s f a c t i o n of C o n d i t i o n (2.1) f o r s m a l l / 3 is r e l a t e d to the f a c t t h a t
f o r U(t) ~ oo, t E T v, t h e c o n d i t i o n a l p r o b a b i l i t i e s of b o t h v a l u e s of x ~ X p r o v e to b e c l o s e t o t h e i r p r o b a b i l i t i e s in the c a s e of an i d e a l g a s . F o r # c l o s e to -¢o the c o n d i t i o n a l p r o b a b i l i t y t h a t x = 0 is c l o s e to 1,
a n d f o r it c l o s e to +oo it is c l o s e to 0 u n i f o r m l y w i t h r e s p e c t to a l l c o n d i t i o n s . If U(t) c a n a c q u i r e i n f i n i t e
v a l u e s and U(t)x(t) = ¢o f o r a c e r t a i n t, then f o r a l l i t a n d fl the c o n d i t i o n a l p r o b a b i l i t y t h a t x = 0 is e q u a l to
1, and the r e a s o n i n g i s p e r f o r m e d o n l y in t h e c a s e of s u f f i c i e n t l y l a r g e n e g a t i v e #. T y p i c a l r a n g e s of the
v a l u e s (fl, #) f o r w h i c h c o n d i t i o n s (2.1) a r e s a t i s f i e d h a v e b e e n s h a d e d in F i g s . 1 and 2. In F i g . 1 we t o o k
into a c c o u n t the f a c t t h a t w h e n x(t) is r e p l a c e d w i t h x'(t) = 1 - x ( t ) a n d it i s r e p l a c e d with p' ----- - ~ - I - ~, U (t)
tETv
t h e c o n d i t i o n a l p r o b a b i l i t y q{0}(0/x(t)) b e c o m e s q{0}(1/x'(t)); t h e r e f o r e , t h e r e g i o n in w h i c h C o n d i t i o n (2.1)
i s s a t i s f i e d c a n b e a s s u m e d s y m m e t r i c a l r e l a t i v e to the l i n e
(2.9)
t
lET v
Note t h a t a r e s u l t a n a l o g o u s to T h e o r e m 1 c a n b e p r o v e d b y t h e R u e l l e - M i n l o s m e t h o d ( s e e [3]). Note a l s o
t h a t f r o m the r e s u l t s o f [2] it is p o s s i b l e to e x t r a c t e x p l i c i t e s t i m a t e s f o r ~ v ( d ) a n d ~ ( d ) e v e n in the c a s e
of nonfinite potentials.
3. W h e r e a s in t h e m u l t i d i m e n s i o n a l c a s e u n i q u e n e s s is p r o v e d o n l y f o r a c e r t a i n s u b r a n g e of the
v a l u e s of t h e p a r a m e t e r s (fl, it), in the o n e - d i m e n s i o n a l c a s e v = 1, u n i q u e n e s s u s u a l l y a p p l i e s f o r a l l (fl, it)
i f the p o t e n t i a l d e c r e a s e s s u f f i c i e n t l y r a p i d l y ; t h i s c o r r e s p o n d s to t h e p h y s i c a l c o n c e p t t h a t p h a s e t r a n s i t i o n s a r e a b s e n t in o n e - d i m e n s i o n a l s y s t e m s .
T H E O R E M 2. A s s u m e v = 1. A s s u m e t h e r e e x i s t s a s e q u e n c e of n u m b e r s p k < 1, k = 0, 1 . . . . .
t h a t Pk "" 0 f o r k --- ~ a n d
i
sup
sup
-~,
I qto,.I (x~ ..... x.+~/x (t))
n=o,1,,,
such
x(O, "x{t):xiO~'x (t), --k~.t<o, n<t 2 x~EX,...,Xn+xE X
- - q[0.,l (x, . . . . .
x,+l/;(t))l~,ok, k =:0, 1. . . . .
(3.1)
where the upper bound is chosen over all n = 0, 1 . . . .
and over all p a i r s of functions x(t), ~(t), t 6 T i \ [ 0 , n ] ,
such that x(t) = ~'(t), if k -< t < 0 and n < t. Then there exists only one Gibbsian distribution. For this d i s tribution the condition of uniform strong mixing is satisfied; this condition indicates the existence of a
function x(d) --* 0 for d -~ ¢o, such that for all finite V CA T I, ~¢ C T 1
304
sup
AE~t.,B6~ y
I Pr { A / B }
Pr {B} J< 7. (d (V, ?))
--
(3 .2)
(see [1], s e c t i o n 5); f u r t h e r m o r e , if the potential is finite, it is p o s s i b l e to p l a c e x(d) = e -°~d, a >0.
the s u m
Y, lU(OIItl< =,
If
(3.3)
t:ltl>-d
then Condition (3.1) is s a t i s f i e d for all (fl,/~).
Note that the p r o p e r t y of u n i f o r m s t r o n g m i x i n g (3.2) b r i n g s with it the p r o p e r t i e s of u n i f o r m r e g u l a r i t y f r o m within and f r o m without.
P r o o f . The b a s i c p o s t u l a t e of the f o r m u l a t e d t h e o r e m is a p a r t i c u l a r c a s e of a m o r e g e n e r a l t h e o r e m
that w a s p r o v e d in detail in [2]. In the c a s e of a finite potential it is d e r i v e d s i m p l y f r o m the e r g o d i c
t h e o r e m f o r M a r k o v c h a i n s . H e r e we will m e r e l y v e r i f y the fact that f r o m Condition (3.3) we obtain s a t i s f a c t i o n of Condition (3.1). F o r this p u r p o s e we note that f r o m (1.3) it follows that if x(t) = ~ ( t ) , - k -<t < 0,
n < t, then the r a t i o
q[..-I (xl . . . . . x.+l/x (t))
qto..., (*, .....
x,,+, '7 (0)
U(s- s)ll
1
s<--k
>exp/--2~
~
ItllU(OI}.•
(3.4)
Itl>/~
Therefore,
-Yi
I qto.,q(xt . . . . . x . + j x ( t ) ) - qt,,..1 (x, . . . . . x~ ~d.~-(t)) l
~,,
xtEX,...,XnTxEX
-- 1--
min (q£o.n](xl . . . . . x,~+t/x (t)), qto.n](xi . . . . . xn+l!x(t)))
~.j
xt EX,...,Xn.t.IEX
-~ 1-- exp {--2,~ ~ l l J j U ( t ) J } x
~
O[o.nl(xl,. . . . . xn..~/x'(l))
~EX,...,XnT1EX
It I > k
= l - - e x p {--2~ ~]
ttl>k
ItllU(OI};
(3.5)
the r i g h t side in this inequality tends to z e r o f o r k --- ~o. It is l e s s than 1 f o r all k if JU(t) J < ~o f o r all
t E TV; this p r o v e s s a t i s f a c t i o n of Condition (3.1) in this c a s e . In the g e n e r a l c a s e , when only (3.3) is
v a l i d , it is still n e c e s s a r y to c h e c k w h e t h e r the r i g h t side in (3.5) is l e s s than 1. P r o c e e d i n g in a m a n n e r
analogous to that u s e d in obtaining the e s t i m a t e (3.5), it is d e m o n s t r a t e d that f o r x(t) - ~ ( t ) , t > n, we have
t
-£
Y
I qto,n](x, . . . . . x,+x/x (t)) - qto.,q(x, ..., x,+dx(t)) I
xtEX,...,xn+IEX
>exp{--2~
~ JU(t)Jltl}
ltl >d
~,
q[o,.](0 ..... O.x,+, .....
x.+jx(t)).
(3.6)
XdTlEX,...,Xa+xEX
F u r t h e r m o r e (see (1.3), (1.4)):
% . 1 (o.
o, xd+l . . . . . x.+l /~(t))
Xd+tEX,.. ,Xn+tEX
i
-- .,~
'~,
%.,,1 (0 . . . . . O, xa+ . . . . . . x,,+~/;(t))
x,EX,...,xn+tEX
>~
,hi.
%'"1(° . . . . o,.~+, ..... x.+jx(o)
Y
qto.l(x, .....~.+,/;(0)
2 x,Ex..-.xn+tex
U[o.n] (xI. . . . . xn+~/'~(t)) x,ex.....xn~,_~ex '
> ~exp/2"[ --f~(ll~ld+linfU(t)l,
er~
d (d-t)2
d ,er,'
~' rain (U (t), 0))}.
(3.7)
Since the r i g h t side of Inequality (3.7) is p o s i t i v e and independent of n and ~(t), it follows f r o m (3.7) and
(3.6) that the c o e f f i c i e n t s Pk in the f o r m u l a t i o n of the t h e o r e m c a n be m a d e l e s s than 1; this p r o v e s s a t i s f a c t i o n of Condition (3.1) w h e n (3.3) is valid.
305
a ,I
r~
F/A~
--
ljl
F/~A
|
I
,
l~1l
I II
~
I
-j--i
P
F i g . 1. R e g i o n i n w h i c h
C o n d i t i o n (2.1) is s a t i s fied. The potential
U(t) ; ~ ¢ o , t E T v .
?"
-........
~
y
F i g . 2. R e g i o n w h e r e
C o n d i t i o n {2.1) i s s a t i s f i e d . T h e p o t e n t i a l U(t)
can acquire the value
-boo.
F i g . 3. C o n t o u r s in
the c a s e ~(t)=x0(t) f o r
the I s i n g m o d e l with
attraction. The arr o w s i s o l a t e the d i r e c tion of contour rotation.
In the c a s e of finite U(O t h e r e s u l t s g i v e n in T h e o r e m 2 c a n b e o b t a i n e d f r o m known e x p l i c i t f o r m u l a s
f o r t h e c o r r e l a t i o n f u n c t i o n s ; it i s new f o r n o n f i n i t e U(t).* T h e q u e s t i o n a s to w h e t h e r C o n d i t i o n {3.3) c a n
b e r e p l a c e d w i t h the w e a k e r c o n d i t i o n (1.1) r e m a i n s open. (On this s u b j e c t s e e the d i s c u s s i o n in [2], s e c t i o n 2.1
4. We w i l l now p r o c e e d to e x a m p l e s o f n o n u n i q u e n e s s of t h e G i b b s i a n d i s t r i b u t i o n v > 1. F i r s t w e
w i l l s t u d y the I s i n g m o d e l with a t t r a c t i o n in w h i c h
U(t) : { a < 0 ,
o,
if
Ill--l,
if
[tl4=l.
(4.1)
We find t h a t f o r s u c h a p o t e n t i a l t h e r e e x i s t at l e a s t two d i f f e r e n t G i b b s i a n d i s t r i b u t i o n s f o r ( s e e (2.9))
F = ~ = va,
~ ~ - - a-lsv,
(4.2)
w h e r e t h e c o n s t a n t su, w h i c h d e p e n d s o n l y on the d i m e n s i o n a l i t y u, w i l l b e d e f i n e d b e l o w d u r i n g the c a l c u l a t i o n s . T h i s f a c t w a s f i r s t d e r i v e d b y R. A. M i n l o s and Ya. G. S i n a i ( o r a l c o m m u n i c a t i o n ) on the b a s i s of
t h e m e t h o d d e v e l o p e d in [4] b a s e d on e q u a t i o n s f o r c o n t o u r s and p r o b a b i l i t y e s t i m a t e s o f a c o n t o u r ( s e e
L e m m a 1 - P e i e r l s [10], G r i f f i t h s [5], and R. L . D o b r u s h i n [6]). The c o n c e p t s d e v e l o p e d in t h i s p a p e r m a k e
i t p o s s i b l e to o b t a i n t h i s r e s u l t in a c o n s i d e r a b l y s i m p l e r m a n n e r a s a d i r e c t c o r o l l a r y o f L e m m a 1.
A l l f o r m u l a t i o n s in t h i s a n d s u b s e q u e n t s e c t i o n s a r e a n a l o g o u s f o r a n y d i m e n s i o n a l i t y v -> 2. F o r
p u r p o s e s o f g e o m e t r i c c o n v e n i e n c e we w i l l a s s u m e , h o w e v e r , t h a t u = 2 (the m u l t i d i m e n s i o n a l g e n e r a l i z a t i o n h a s b e e n s t u d i e d in d e t a i l in [6]). B e l o w w e s h a l l s t u d y j u s t two c a s e s o f t h e b o u n d a r y c o n d i t i o n s :
x(t) = [ x°(t) ::- O, t E T 2 ~ V ,
(4.3)
[x~(t)= 1, tET~\V,
a n d s h a l l a s s u m e t h a t V = {t1. . . . .
t [ v l } is a c e r t a i n s q u a r e . We s h a l l a s s o c i a t e e a c h p o i n t t E T 2 of the
t w o - d i m e n s i o n a l l a t t i c e with a s q u a r e h a v i n g s i d e s o f l e n g t h 1 p a r a l l e l to the c o o r d i n a t e a x e s w i t h i t s c e n t e r a t t h a t p o i n t . W e w i l l a l s o d e s i g n a t e it with t h e s y m b o l t. E a c h f u n c t i o n ~(t), t E T2,-~(t) = 0, 1, i s
j u x t a p o s e d w i t h a g e o m e t r i c p a t t e r n {see F i g . 3) t h a t i s o b t a i n e d if we s h a d e the s q u a r e s for w h i c h E(t) = 1.
We w i l l d e f i n e a b o u n d a r y s i d e a s t h e s i d e of a s q u a r e t h a t s e p a r a t e s s h a d e d and u n s h a d e d s q u a r e s . G e o m e t r i c a l l y it is o b v i o u s t h a t an e v e n (0, 2, o r 4) n u m b e r of b o u n d a r y s i d e s e m a n a t e f r o m e a c h v e r t e x of
t h e s q u a r e s , and t h a t e a c h e n s e m b l e of s i d e s o f s q u a r e s w i t h i n V h a v i n g t h a t p r o p e r t y c o r r e s p o n d s to a
c e r t a i n f u n c t i o n ~(t); f u r t h e r m o r e , f o r a s p e c i f i e d b o u n d a r y f u n c t i o n x(t) t h i s a m o u n t s to a o n e - t o - o n e
r e l a t i o n s h i p if x(t) = x ( t ) , t E T 2 ~ V . We w i l l d e f i n e t h e l e n g t h of t h e s i t u a t i o n b o u n d a r y (x 1. . . . . x i v I ) ,
x i EX, i = 1 . . . . .
IV], f o r a s p e c i f i e d f u n c t i o n x(t) a s t h e n u m b e r r ( x l , . . . , x I v I / x ( t ) ) , e q u a l to the s u m
o f the n u m b e r of p a i r s of p o i n t s t i , tj, i ~ j , such t h a t I t i - t j ] = l , xi ~ x j , and the n u m b e r of p a i r s of p o i n t s
t i, t .E T 2 \ V , s u c h t h a t I t i - t l = 1, xi ~ x(t). It c a n b e s e e n t h a t l~(xl . . . . .
x l v I / x ( t ) ) is the o v e r - a l l n u m b e r
* A f t e r t h i s p a p e r h a d b e e n s u b m i t t e d f o r p u b l i c a t i o n , the a u t h o r r e c e i v e d a p r e p r i n t o f a p a p e r b y R u e l l e [9]
in w h i c h a r e s u l t c l o s e to t h a t c o n t a i n e d in T h e o r e m 2 w a s p r o v e d .
306
i
!
i il~l
I i
EL
II
I
f
L ........
i
I ]
I
I
I [cZ~dJ
I
.-z,. ~
.;,
I,
L J _ . L . L J _ L .;
Fig. 5. R e g i o n s of u n i q u e n e s s and n o n u n i q u e n e s s
in the I s i n g m o d e l with
a t t r a c t i o n . The h y p o t h e t i c a l continuation of the
n o n u n i q u e n e s s line is
shown d a s h e d .
Fig. 4. R e s u l t of a p plying the t r a n s f o r m ation T G to the s i t u a t i o n in Fig. 3, w h e r e G
is the c o n t o u r of m a x i m u m length.
"~',.'~'~,'~ ~ @ ~ ~ ,
!F~ I
,E~_(_~ _ ~, _ ~ _ ~ 4 _ B . .
Fig. 6. C o n t o u r s for
the Ising m o d e l with
repulsion.
of b o u n d a r y s i d e s f o r the f u n c t i o n x ( t ) = x(t), t E T 2 \ V , ~(t i) = x i, ti E V. The s i t u a t i o n b o u n d a r y (xl . . . . .
xl VI ) is defined as the e n s e m b l e of b o u n d a r y sides f o r ~(t). Note that f r o m (1.3) we obtain the following
r e s u l t f o r # = ~ = 2a:
~a
exp [ 7 r (xl . . . . .
qv (x, . . . . .
xwl,'x(t)) }
(4.4)
xwl 'x (t))
exp { ~ r ( x , . . . . . xwr,'x(0) }
xLf:X....,xlvI(:X
G e o m e t r i c a l l y it is e v i d e n t that the e n s e m b l e of all b o u n d a r y s i d e s c a n be subdivided i n t o n o n - s e l f - i n t e r s e c t i n g c l o s e d b r o k e n lines t h a t we will call c o n t o u r s (Fig. 3 shows eight c o n t o u r s ) ; f u r t h e r m o r e , it is
c o n v e n i e n t to a s s u m e t h a t all c o n t o u r s p e r f o r m t u r n s at v e r t i c e s w h e r e four b o u n d a r y s i d e s c o n v e r g e .
U n d e r t h e s e conditions two c o n t o u r s e i t h e r do not i n t e r s e c t , o r one of t h e m is i n s e r t e d within the o t h e r .
(The a m b i g u i t y of the s u b d i v i s i o n of a b o u n d a r y into c o n t o u r s is insignificant f o r o u r p u r p o s e s . )
LEMMA 1. F o r a Gibbsian d i s t r i b u t i o n in the s q u a r e V with b o u n d a r y conditions of the type (4.3) the
p r o b a b i l i t y Pr{G} that the situation (x 1. . . . . x IV I) c o n t a i n s a c e r t a i n c o n t o u r G (i.e., m o r e o r e c i s e l y , that
e a c h of the sides of the c o n t o u r G is included in the situation b o u n d a r y (x 1. . . . . xl VI)) is s u c h that
Pr (6} < exp ( ~ 1 6 ] } ,
(4.5)
w h e r e IGi is the length of the c o n t o u r G.
This l e m m a was p r o v e d in [5] and [6]. We will give a s i m p l e p r o o f of it, since it will be g e n e r a l i z e d
in the subsequent s e c t i o n s . A s s u m e ~(G) is an e n s e m b l e of situations (x 1. . . . .
x l v I) c o n t a i n i n g the c o n t o u r
G. T h e n f r o m (4.4) it follows that
z;
%"
exp / ~ r (x,,
Pr {G} := (*..... *lVt)e'9(a)
/,
exp
x,(~x,...,XlVle.x"
r (xx . . . .
.,Xwl/X(t)) }
(4.6)
Xlvl/X (t))
We i n t r o d u c e the t r a n s f o r m a t i o n TG that j u x t a p o s e s e a c h s i t u a t i o n (x1 . . . . .
following rule:
xi ::: xi,
if
.x~'i::= 1 - - Xi, if
x lv I) E ~(g) a c c o r d i n g to the
t~ does not lie outside the contour G,
ti lies inside the contour G
(4.7)
(Fig. 4). In o t h e r w o r d s , the t r a n s f o r m a t i o n T G annihilates the c o n t o u r G and does not c h a n g e o t h e r c o n t o u r s . It is evident that
F (x t . . . . . :~'ivi/x (t)) := r (x, . . . . . xlvl/x (t)) -- ] G [
(4.8)
307
and that d i f f e r e n t s i t u a t i o n s (x, . . . . .
x l v I) y i e l d d i f f e r e n t s i t u a t i o n s (~f . . . . .
xjvl).
Therefore,
F (xl . . . . . xWl'x (~))
exp
"
= e x P ] 2 I GI '
I
(4.9)
-I
which p r o v e s the l e m m a .
Note that if the point to = (to1, t02) 6 T 2 lies inside the c o n t o u r G and this c o n t o u r c o n t a i n s a side of a
s q u a r e with its c e n t e r at t = (t 1, t2), inside G, then the length of the c o n t o u r G m u s t be no l e s s than 2(]t~-tt[ +
I t02-t~l + 2). F u r t h e r m o r e , no m o r e than 3 d - ! b r o k e n lines d p a s s t h r o u g h the given side. A s s u m e rr(t0) is
the p r o b a b i l i t y that the point to lies inside at l e a s t one of the c o n t o u r s s p e c i f i e d by the situation (x t . . . . .
x[ V[ ). S u m m i n g o v e r all c o n t o u r s G that contain t o inside t h e m and p a s s along a side of the s q u a r e t, and
taking into a c c o u n t the fact that u n d e r these conditions a c o n t o u r of length d will be counted d t i m e s and that
the length d of a c o n t o u r is a l w a y s a multiple of 2 and each s q u a r e has four s i d e s , we obtain the following
r e s u l t f r o m (4.5):
a ( t o ) < 4 ~,
I t
~
s : ~ e x p {(~a + 21n3)k}
2
2 co
4
exp {[~aq- 2 In 3}
= ~ ~', k ~ l k ""~ ( k - - l)2]exp {(~a -I- 2 In 3) k} ~ ~'(t--exp {~a + 21n 3}F'
if exp {/3a + 2 In 3} < 1. A s s u m e s 2 > 0 is the s o l u t i o n of the equation
4
exp (-- s~ + 2 In 3}
3 (l -- exp {-- s~ -}- 2 In 3})3
I
'2
(4.10)
Then, if Condition (4.2) is valid, it follows that ~(t0) -<3/< 1/2 f o r all t o E T v. But if x(0 = x0(t) and the point
to lies outside any c o n t o u r , then the field value is ~(t0) = 0. F r o m this it follows that the following r e l a tionship is valid f o r a Gibbsian d i s t r i b u t i o n with the b o u n d a r y conditions x0(t) :
Pr (~ (ti) == 1} < T ~ ~i ,
ti O-V.
(4.11)
It is obvious that Condition (4.11) i s o l a t e s a c l o s e d s e t of d i s t r i b u t i o n s in the m e t r i c i n t r o d u c e d in [1] (see
[1], s e c t i o n 2). T h e r e f o r e , in view of c o m p a c t n e s s (see [1]) and T h e o r e m 1 f r o m [1] t h e r e e x i s t s a Gibbs i a n field ~(t) s u c h that
Pr {~-(t) = I } ~ T ~
t,
tO_T".
(4.12)
Analogous r e a s o n i n g b a s e d on the b o u n d a r y conditions x(t) = xi(t) l e a d s to the c o n c l u s i o n that a Gibbsian
field "~(t) e x i s t s s u c h that
Pr {~(t) = 1} ~ 1 - - q ' ~ , ,
to-T ~,
(4.13)
.a
and that the Gibbsian fields ~(t) and ~"(0 c a n n o t c o i n c i d e . Taking into a c c o u n t the fact that Condition (4.11)
is invariant r e l a t i v e to shifts and applying r e a s o n i n g that is a n a l o g o u s to that u s e d in d e r i v i n g T h e o r e m 1
f r o m [1], we a l s o find that noncoinciding t r a n s l a t i o n - i n v a r i a n t d i s t r i b u t i o n s e x i s t f o r which (4.12) and
(4.13) a r e valid, r e s p e c t i v e l y . T h e s e two d i s t r i b u t i o n s a r e obtained f r o m one a n o t h e r b y r e p l a c i n g 0 with
1 and 1 with 0.
The m e t h o d o f c o r r e l a t i o n functions f o r M i n l o s - S i n a i c o n t o u r s [4] can be u s e d to p r o v e that the
f o r m u l a t e d d i s t r i b u t i o n s a r e the l i m i t s of the Gibbsian d i s t r i b u t i o n s q v i ( " /x0(t)) and q v i ( " /xl(t)), r e s p e c t i v e l y , f o r a n y expanding s e q u e n c e of c u b e s Vi. T h e i r m e t h o d a l s o m a k e s it p o s s i b l e to Drove the
u n i q u e n e s s of the Gibbsian d i s t r i b u t i o n f o r 13 > rio, w h e r e 13o < ~o is a c e r t a i n constant, and f o r a l l / z ~#.
In Fig. 5 the r e g i o n for which u n i q u e n e s s of the Gibbsian d i s t r i b u t i o n is p r o v e d has b e e n shaded, and the
line on which, as d e m o n s t r a t e d , t h e r e is no u n i q u e n e s s is d e s i g n a t e d b y a thick line.
308
~UA~V/A'
V/AV/~VA~
~
VA_~_
VAINN
VA~V~i~
_VA_W2_{~ J
F i g . 7. R e s u l t of a p p l y i n g the t r a n s f o r m a t i o n TG to the s i t u a t i o n
in F i g . 6, w h e r e G is t h e
c o n t o u r of m a x i m u m
length.
T h e n a t u r a l h y p o t h e s i s c o n s i s t s of t h e f a c t that the G i b b s i a n d i s t r i b u t i o n
i s u n i q u e f o r a l l {~,/3), w i t h the e x c e p t i o n of p a i r s (p, fl, w h e r e / 3 > c 2 ( - a - i ) and c 2 i s
t h e v a l u e f o u n d b y O n s a g e r ( s e e [7]). T h e i m p o r t a n t q u e s t i o n of w h e t h e r the c l a s s of
G i b b s i a n d i s t r i b u t i o n s is e x h a u s t e d b y the two d i s t r i b u t i o n s found a b o v e and t h e i r
l i n e a r c o m b i n a t i o n s a l s o r e m a i n s open. The f o l l o w i n g h y p o t h e s e s c a n b e a d v a n c e d .
T h e r e a r e c e r t a i n c o n c l u s i o n s s u b s t a n t i a t i n g t h e p r o p o s i t i o n t h a t t h i s is so f o r
v = 2. H o w e v e r , f o r v> 2 t h i s i s e v i d e n t l y no l o n g e r s o . It is p o s s i b l e to a s s u m e t h a t
f o r v = 3 t h e r e is an e n t i r e a d d i t i o n a l f a m i l y of e x t r e m e p o i n t s in t h e e n s e m b l e of
G i b b s i a n d i s t r i b u t i o n s t h a t a r e l i m i t i n g for d i s t r i b u t i o n s with b o u n d a r y c o n d i t i o n s x(t), t E T 3, s u c h t h a t x(t i, t 2, t 3) = 1 f o r t i -> c and x(t 1, t 2, t 3) = 0 f o r
t 1 < c, a s w e l l a s a n a l o g o u s f a m i l i e s t h a t a r e o b t a i n e d w h e n t 1 is r e p l a c e d b y t 2
and t3; i t is a s s u m e d t h a t t h e s e e x h a u s t a l l t h e e x t r e m e p o i n t s . Such d i s t r i b u t i o n s c o u l d d e s c r i b e s u r f a c e p h e n o m e n a on the b o u n d a r y of the p h a s e s .
A l l t h a t w a s s a i d a b o v e in c o n n e c t i o n with t h e I s i n g m o d e l w i t h a t t r a c tion is not d i f f i c u l t to e x t e n d to the m o r e g e n e r a l c a s e of a r b i t r a r y p o t e n t i a l s
U(t) f o r w h i c h t h e n e g a t i v e p a r t is in a c e r t a i n s e n s e ( p r e c i s e l y d e f i n e d in a
p r e v i o u s p a p e r b y t h e a u t h o r [8]) g r e a t e r t h a n the p o s i t i v e p a r t . In [8] it w a s p r o v e d t h a t f o r s u c h p o t e n t i a l s and f o r s u f f i c i e n t l y l a r g e fl a n e s t i m a t e a n a l o g o u s to the e s t i m a t e in L e m m a 1 is v a l i d .
5. We w i l l s t u d y the I s i n g m o d e l with r e p u l s i o n , w h e r e
V(t)={aDO'
~f
o, ~f
{tl= 1,
1t{4=I
(5.1)
W e find that for such a potential at least two different Gibbsian distributions exist for a > I~I/v,
(5.2)
T h i s r e s u l t is e v i d e n t l y new f r o m the m a t h e m a t i c a l s t a n d p o i n t .
We s h a l l s t u d y t h e g e o m e t r i c i n t e r p r e t a t i o n i n t r o d u c e d in t h e p r e v i o u s s e c t i o n , b u t we s h a l l now d e fine a b o u n d a r y s i d e a s a n y s i d e t h a t s e p a r a t e s two s h a d e d o r two u n s h a d e d s q u a r e s . C o r r e s p o n d i n g l y , a
l o n g s i t u a t i o n b o u n d a r y (x! . . . . .
x IV I ) with the b o u n d a r y f u n c t i o n x(t) is d e f i n e d a s t h e n u m b e r r ( x I . . . . .
x I v i / x ( t ) ) , e q u a l to the s u m o f the n u m b e r o f p a i r s o f p o i n t s t i , t j , { ~ j , s u c h t h a t I t i - t j [ = 1, x i = x j , and
t h e n u m b e r o f p a i r s of p o i n t s ti E V, t E T 2 \ V , s u c h t h a t ] t i - t ] = 1, x i = x(t). The o b v i o u s m e a n i n g of
"F(x I . . . . .
x i v I / x ( t ) ) is the s a m e a s it is in the p r e v i o u s e x a m p l e . F r o m (1.2) it f o l l o w s that
)}
exp 1-( ~ - ~ v
~ x~+ ~o P (x,..... xlvL/X(0)
qv(X~ . . . . . x'vl/x(t)) =
t
' ~ ' _ iv,
x,EX....,x{v}EX
.
(5.3)
t
This formula can easily be obtained from (4.4) and from the fact that the s u m r(x{ ..... x{ v{/x(t)) +
~(x i..... x{v{/x(t)) is independent of (x! ..... x{vl).
W e will call the point t -- (tI, t2) even if {tI + t2{ is an even number, and odd if {tI + t2{ is an odd n u m ber. It is obvious that separation into even and odd squares corresponds to the separation of a chessboard
into white and black squares. In our subsequent analysis w e will a s s u m e that x(t) acquires the values x0(t)
or xi(t), w h e r e
x 0 (t) : : { O, t • T2~V,
1, t (~~ \ V ,
Xt(0 : :
if
if
t even,
t odd.
0, t E T ~ \ V, ff
t odd,
1, t E T 2 ~ V ,
t even.
if
(5.4)
G e o m e t r i c a l l y i t is o b v i o u s t h a t in t h e s e c a s e s the e n s e m b l e of a l l s i d e s h a s the s a m e p r o p e r t i e s a s it d o e s
in S e c t i o n 4 o f t h i s p a p e r ; in p a r t i c u l a r , it c a n b e s p l i t up into c o n t o u r s ( F i g . 6).
309
L E MMA 2. For,"a G i b b s i a n d i s t r i b u t i o n i n the s q u a r e .V w i t h b o u n d a r y conditions, o f , t h e . f o r m (5;4)
t h e p r o b a b i l i t y Pr{G} t h a t the a r r a n g e m e n t (x 1. . . . . . xiVI) c o n t a i n s a c e r t a i n c o n t o u r G is s u c h t h a t
w h e r e [ GI is the l e n g t h of t h e c o n t o u r .
T h e p r o o f is p e r f o r m e d in a m a n n e r a n a l o g o u s to the p r o o f o f L e m m a 1. In p a r t i c u l a r , p r o c e e d i n g
a n a l o g o u s l y to {4.6), we u s e (5.3) to obtain.
_
{-
+-7.1,
Pr {G} = (x......ZWli~(o)
. . . . .
,=l
f
r
.~
IVl
(5.6)
<,,>)}
a
. . . . .
x,.EX,...,Xl vlE X
T h e p r i n c i p a l d i f f e r e n c e l i e s in the f a c t t h a t i n s t e a d of a t r a n s f o r m a t i o n T G we i n t r o d u c e a d i f f e r e n t t r a n s f o r m a t i o n ~ G ( x 1. . . . .
xl VI) = (~l . . . . .
~ I V I ) , w h e r e (x 1. . . .
xl VI) E ~(G); we d e f i n e t h i s t r a n s f o r m a t i o n
( F i g . 7) a c c o r d i n g to the f o l l o w i n g r u l e . A s s u m e t j i is a s q u a r e t h a t is d i r e c t l y b e l o w the s q u a r e ti. T h e n
-'~i ~ xi, if
xi ~ xl~, if ti
ti lies outside the contour G,
and
[;]i tie inside the contour G,
x t = 1 - - Xir if
(5.7)
tt lies inside G, aLadrji lies outside G.
In o t h e r w o r d s , it c a n b e s t a t e d that the t r a n s f o r m a t i o n ~ G a n n i h i l a t e s the c o n t o u r G, d o e s n o t c h a n g e c o n t o u r s l y i n g o u t s i d e G, and r a i s e s a l l c o n t o u r s l y i n g i n s i d e G u p w a r d b y one ( s e e F i g . 7). U n d e r t h e s e c o n ditions
_
F(XL. . . . . ~'IVl/X(t)) = F(x~ . . . . .
xinlx(t))--lG I,
IVl _
WI
i
i=l
I
~ x , - - ~ , x , < Ialh°r
2
(5.8)
'
w h e r e I G I h o r is the n u m b e r of h o r i z o n t a l s i d e s of the c o n t o u r G. C o n c e p t s a n a l o g o u s to (4.9) now m a k e it
p o s s i b l e to d e r i v e t h e f o l l o w i n g r e s u l t f r o m (5.6):
A n a n a l o g o u s e s t i m a t e is t r u e , o f c o u r s e , ff the n u m b e r o f h o r i z o n t a l s i d e s o f the c o n t o u r G i s r e p l a c e d
with t h e n u m b e r of v e r t i c a l s i d e s I Giver. S i n c e rain (I G l h o r , IGIver) -< 1/21GI, it f o l l o w s t h a t (5.5) d e r i v e s f r o m (5.9) and t h e a n a l o g o u s i n e q u a l i t y w i t h I G I v e r .
T h e n r e a s o n i n g f u l l y a n a l o g o u s to t h a t p e r f o r m e d in S e c t i o n 4 d e m o n s t r a t e s t h a t ( s e e (4.11)) f o r a
G i b b s i a n d i s t r i b u t i o n with the b o u n d a r y c o n d i t i o n s x0(t) we h a v e
P {~(t,)= I} ~ ' ~ - - t 2 ,
P{~(ti)=0}~T~
i,
if
if
[i is even l i e V,
ti is odd
(5.10)
tt6. V,
and a n a l o g o u s i n e q u a l i t i e s f o r the G i b b s i a n d i s t r i b u t i o n w i t h the b o u n d a r y c o n d i t i o n s xl(t). F r o m t h i s it f o l lows t h a t two n o n c o i n c i d i n g G i b b s i a n d i s t r i b u t i o n s e x i s t . T h e s e two d i s t r i b u t i o n s m a k e the t r a n s i t i o n into
one a n o t h e r when the f i e l d is s h i f t e d b y 1. T h u s , t h e s e two d i s t r i b u t i o n s do not d i f f e r f r o m e a c h o t h e r in
t h e i r m a c r o s c o p i c c h a r a c t e r i s t i c s . The r a n g e s o f v a l u e s of the p a r a m e t e r s (/3,/~) for w h i c h u n i q u e n e s s
and n o n u n i q u e n e s s of the G i b b s i a n d i s t r i b u t i o n , r e s p e c t i v e l y , h a v e b e e n p r o v e n in this c a s e a r e shown in
F i g . 8. The l i n e s e p a r a t i n g the u n i q u e n e s s and n o n u n i q u e n e s s r e g i o n s m u s t p a s s s o m e w h e r e b e t w e e n t h e m ;
t h i s l i n e c a n a l s o b e d e s c r i b e d a s the line f o r m e d by the b r a n c h i n g p o i n t s of the o p e r a t o r Q i n t r o d u c e d in
(2.4). It is n a t u r a l to a s s u m e t h a t at t h o s e b r a n c h i n g o o i n t s a n a l y t i c i t y of the m a c r o s c o D i c f i e l d c h a r a c t e r i s t i c s is v i o l a t e d ; h o w e v e r , t h i s r e m a i n s u n p r o v e d . With r e s p e c t to the c o m p l e t e d e s c r i o t i o n of a l l
G i b b s i a n d i s t r i b u t i o n s we c a n s t a t e a h y p o t h e s i s a n a l o g o u s to the one c i t e d in S e c t i o n 4. If we a d o p t t h i s
h y p o t h e s i s , then w e f i n d t h a t in the i n v e s t i g a t e d e x a m p l e t h e t r a n s l a t i o n - i n v a r i a n t d i s t r i b u t i o n is unique.
310
6. We s h a l l now s t u d y a n y p o t e n t i a l U(t), t E T u, s u c h t h a t
I
7_-_5
....
- -
U(t)::o~,
I
Itl..:i,
~
(6.1)
[U(t)lltt.~oo.
itI i
We find t h a t f o r s u c h a p o t e n t i a l t h e r e e x i s t a t l e a s t two d i f f e r e n t G i b b s i a n
d i s t r i b u t i o n s when
!
~/2~>c,
i
~>
~ ( ":ZTv
*~ - C
T
)-1
(6.2)
s,.,
where
c = ± ma×
F i g . 8. U n i q u e n e s s and
nonuniqueness regions
in the I s i n g m o d e l with
repulsion. Their hypot h e t i c a l b o u n d a r y is
s h o w n b y the d a s h e d l i n e .
V
it;llU(t)I,
Y,
t = (6 . . . .
, t" i.
ll~Tv
j=l,...,v
F o r the p a r t i c u l a r c a s e U(t) = 0 , ] t] ~ 1, t h e a n a l o g o u s r e s u l t w a s p r o v e d b y
m e a n s of a m o r e c o m p l e x m e t h o d b y M o s c o w S t a t e U n i v e r s i t y s t u d e n t Sukhov
in his t h e s i s w r i t t e n u n d e r the s u p e r v i s i o n of Ya. G. S i n a i .
T h e d e r i v a t i o n of t h i s r e s u l t is a n a l o g o u s to the c o n c l u s i o n d r a w n in
S e c t i o n 5. In p a r t i c u l a r , w e a d o p t the d e f i n i t i o n s of the b o u n d a r i e s and the
f u n c t i o n s x0(t), xl(t) i n t r o d u c e d t h e r e . We c a l l the s i t u a t i o n (x I . . . . .
x i v ]) a l l o w e d if no two s h a d e d s q u a r e s
h a v e c o m m o n s i d e s . It is o b v i o u s t h a t in v i e w of (6.1) and (1.2) the p r o b a b i l i t i e s o f n o n a l l o w e d s i t u a t i o n s
a r e e q u a l to z e r o . A s s u m e f o r s i m p l i c i t y t h a t Ivl = / 2 , w h e r e l is an e v e n n u m b e r . It is c l e a r that the
n u m b e r of s h a d e d c e l l s in one c o l u m n of the s q u a r e V is e q u a l to l / 2 - g / 2
f o r an a l l o w e d s i t u a t i o n , w h e r e g
is the n u m b e r of h o r i z o n t a l b o u n d a r y s i d e s in the c o l u m n . The a n a l o g o u s p o s t u l a t e is a l s o v a l i d f o r r o w s ,
to t h a t f o r the a l l o w e d s i t u a t i o n (x 1. . . . . . x J v [)
exp --/3( T r (x,,...,X!Vl,X (t))
qV (xt,'",XlVI/X(t))
F 2
=
f
x~EX,...,XlVlE X
.
-ti}
~=1i=t. iCq
' ~
1
Ivl
!=1 ter~..\v
IVl
(6.3)
l,q
,
"
i=ltErvNV
L E M M A 3. F o r a G i b b s i a n d i s t r i b u t i o n in the s q u a r e V w i t h b o u n d a r y c o n d i t i o n s of the f o r m (5.4)
t h e p r o b a b i l i t y is g i v e n b y the f o r m u l a
T h e p r o o f o f t h a t l e m m a i s p e r f o r m e d a c c o r d i n g to the s a m e t e c h n i q u e a s t h a t u s e d in the p r o o f of
L e m m a s 1 and 2 on the b a s i s o f the t r a n s f o r m a t i o n ~ G i n t r o d u c e d in (5.7). H e r e it s h o u l d b e n o t e d that
t h e t r a n s f o r m a t i o n ~ G c o n v e r t s an a l l o w e d s i t u a t i o n into a n o t h e r a l l o w e d s i t u a t i o n , and a l s o t h a t f o r the
a l l o w e d s i t u a t i o n s we h a v e
/
, ,v
i
-Z•1 IVl
IH
y
i=i i=t. j~i
Ivl
Ivl
~ x,xju ¢,- 6-) _L ~,
~, .~-,x¢) u (t, - t)
i = i t~_T~'\V
j=l,i~=]
lVl
},x~U(t, - #) - ~ ,
I
~ .~,x (t)u(t,-t) <
i= ,tErv\v
I
~,
Iu(t-7)l
(6.5)
t:is insidethe contourG,
t--is outsidethe contourG,
It-tl.~ l
T h e n we c o n n e c t the p o i n t s t and ~ w h e r e t = (t l, t2), ~ = ~ 1 , ~ 2 ) , b y m e a n s of a c o r n e r c o n s i s t i n g of a h o r i z o n t a l s e g m e n t with a l e n g t h I t l - t i I a n d a v e r t i c a l s e g m e n t of l e n g t h I t 2 - ~ l (it is a s s u m e d that the f i r s t
c o o r d i n a t e a x i s is h o r i z o n t a l ) . If t is i n s i d e t h e c o n t o u r G and ~ is o u t s i d e it, then t h i s c o r n e r i n t e r s e c t s
one of the s i d e s o f the c o n t o u r G; f u r t h e r m o r e , f o r f i x e d s = t - ~ = ( s 1, s 2) the g i v e n v e r t i c a l s i d e of the
c o n t o u r G is i n t e r s e c t e d b y no m o r e t h a n Isl~ c o r n e r s c o r r e s p o n d i n g to d i f f e r e n t t, ~ ' p a i r s . We h a v e an
a n a l o g o u s r e s u l t for the h o r i z o n t a l s i d e s of the c o n t o u r G. F r o m t h i s we o b t a i n
311
Iu(t --T)l
~,
t--is inside the contour G,
?--is outside the contour G,
Jt--t[- i
~er',H
~ 1Is, llU(s)l
% 2L,IOlnor
/
+t~ ~ers~r.,,.is,
~-~
lls~/lU(s) l)~ClOl"
(6.6)
Now the postulate of the l e m m a derives f r o m (6.3).
/z
Fig. 9. Regions of uniqueness and nonuniqueness in
the case of a potential of
the type (6.1). T h e i r hypothetical boundary is shown
by the dashed line.
Then it is n e c e s s a r y to repeat what was said in the last p a r t of Section 5, except that the r e f e r e n c e to Fig. 8 must be replaced with a r e f e r enee to Fig. 9. In such a model the phase transition evidently exists for
all ft.
The method developed in Sections 4, 5, and 6 makes it possible to
e s t i m a t e the region of nonuniqueness of the Gibbsian distribution for other
potentials as well.
LITERATURE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
312
CITED
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R . A . Minlos, "Regularity of the limiting Gibbsian distribution," Funktsional'. Analiz i Ego Prilozhen.,
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R.A. Minlos and Ya. G. Sinai, "New results on phase transitions of the first kind in models of a lattice g a s , ' Trudy Mosk. Matem. Obshchestva, 1_!1,213-242 (1967).
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R . L . Dobrushin, 'Existence of a phase transition in the two-dimensional and three-dimensional Ising models," Teor. Veroyatn. i Ee Primen., 1_O.0,No. 2, 209-230 (1965).
K. Huang, Statistical Mechanics, N. Y.-London, John Wiley (1963),
R.L. Dobrushin, Existence of Phase Transition in Models of a Lattice Gas, Proc. Fifth Berkeley
Syrup. on Math. Stat. and Probabilities, Vol. 3, Univ. of Calif. Press (1967), pp. 73-87.
D. Ruelle, Statistical Mechanics of a One-Dimensional Lattice Gas, Centre de Physique Theorique de
l'Ecole Polyteehnique, Preprint (1968).
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