- Email: [email protected]
Rigorous renormalisation group and disordered systems
Physica A 163 (1990) 31-37 North-Holland
RIGOROUS
RENORMALISATION
GROUP AND DISORDERED
SYSTEMS
J. BRICMONT Znstitut de Physique Thkorique, Universitti 1348 Louvain-la-Neuve, Belgium
Catholique
de Louvain,
Chemin du Cyclotron,
2,
A. KUPIAINEN Mathematics
Department,
Rutgers
University,
New Brunswick,
NJ 08540, USA
We consider random walks in a non-symmetric random environment. We report a recent result, based on a Renormalisation Group approach, showing that for d > 2, these walks are diffusive when the disorder is weak.
1. Introduction
A standard way to model the effect of disorder and impurities in physical systems is to replace certain parameters of the system by random variables. Random walks in random environments provide a simple example of such a procedure. A random walk on Zd, with steps taken at unit times, is defined by a set of transition probabilities p(x, y), x, y E Zd, satisfying:
Pk
7
Y) 2
0
3
PC6Y) = 1 .
(1)
If the walk takes place in a homogeneous environment, these probabilities are translation invariant: p(x, y) = p(x - y). To model an inhomogeneous environment, one takes p(x, y) to be random variables themselves. The usual questions about random walks (recurrence, diffusive behaviour, . . .) are now to be answered for almost every set of transition probabilities. In this talk we shall sketch how rigorous Renormalisation Group methods can be used to prove that, above two dimensions, random walks in a “weakly” random environment diffuse just as ordinary random walks [2]. This is to be contrasted with the results of Sinai’ [l] who showed that a similar walk in one dimension moves, in a time t, only over a distance of order (log t)*. 0378-4371/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland)
32
J. Bricmont and A. Kupiainen
I Rigorous renormalisation group and disordered systems
Our RG analysis bears some resemblance with the one used before, but in a totally different context, to show that a small random field does not destroy the phase transition in the Ising model, above two dimensions [3]. However, in the Ising model, we dealt with equilibrium phenomena while here we deal with a non-equilibrium system. In either case, however, we do not, in order to apply the RG analysis, first reduce the model to a pure, non-disordered one, as is often done. Rather, we define in a some sense two RG flows, one on the statistical system itself (the random walks, or the Ising spins) and another one on the disorder (the environment or the random fields). We must emphasise that the fixed points towards which the RG flow converges are quite trivial, both for the Ising model and for the random walk. The extension of this method to non-trivial fixed points is an open problem.
2. Random walks in random environments Here we shall define more precisely the model, state our results and recall some previously known facts. We consider nearest-neighbour transition probabilities on Hd: (2) where if Ix-y]
= 1
otherwise defines the usual random walk and {b(x, y)]]x - y( = l} are random variables satisfying the following properties:
4
b) w-5 Y>, be’, Y') are independent 4 b(x, , _= .s2 (small), d) Prob
(
for x # x’ and identically distributed,
& + b(x, y) < eeN
In c) and below the bar denotes the average over the b-distribution. Condition a) means that the p(x, y) are transition probabilities, i.e. satisfy (1). Condition b) is in some sense what makes the problem difficult: there is no
1. Bricmont and A. Kupiainen
I Rigorous renormalisation group and disordered systems
33
relation between p( y, X) and p(x, y), i.e. between crossing a bond in one direction and in the opposite direction. This, in turn, may cause “trapping”: for example, we may have a nearest neighbour pair (w, z) with p(w, z) and p(z, w) almost equal to one and p(w, w’), w’ # z, p(z, z’), z’ # w almost zero. Then, once the walk reaches either w or z, it may be stuck in between these two sites for a long time. Since nothing forces p(w’, w), w’ # z or p(z’, z), z’ # w to be small, the walk can easily reach z or w. It would not be too bad if this was the only example. However, it is easy to see that this pattern occurs on all scales: even if each individual b(x, y) is small, they may be (un)suitably oriented in such a way as to “push” the walk and confine it, for a long time, in a given region. Of course, the walk cannot be trapped forever: this is insured by conditions d) and b): d) says that the probability that’some p(z, w) is almost one (which, by the sum rule (l), means that some p(z, z’) is small) is unlikely. Coming back to our first example, if p(z, z’) = emN, for all z’ # w, then the time it takes to escape from the trap is eN, which is not too large, ___ given the ~small density (=ePNis) of such events. Condition b) implies that b(x, y) = b(x, y’) and, by a), that b(x, y) = 0. So there is no net drift in the system. This, in turn, will imply that the large scale traps described above are even more unlikely than those involving only two sites. Finally, condition c) means that typical b(x, y)‘s are small. Under the above hypotheses, we show the following: define the mean square displacement as usual, t-1
(x2(t))=
c x2(t)lgoP(X(9?x(i + 1)) (x(i)):=1
(3)
with x(O) = 0. Then, if d > 2, lim t-e-
0
=
D(E)
t
with probability
one. Moreover,
II(&) = D(0) +
O(2).
(4)
‘So the behaviour is diffusive, but with a “renormalized” diffusion constant. One of the first results on random environments is due to Sinai [l] who considered the same model, but in one dimension. There, he proved, with no restriction on E, that (x’(t))
= (ln t)” .
34
J. Bricmont
and A. Kupiainen
I Rigorous renormalisation group and disordered
systems
So, it was natural to ask whether such a subdiffusive behaviour might also occur in higher dimensions. Using a renormalisation group calculation, Fisher [4] and Derrida and Luck [S, 61 concluded that, for d 2 2, normal, diffusive behaviour takes place. Strictly speaking, this calculation applies only to small E, but Fisher [4] argues that the same result should hold for all E. This conclusion is at variance with the numerical results of [7], which concern d = 2 and strong disorder (large E). Moreover, Durrett [8] constructed counterexamples (but with dependent transition probabilities, i.e. which did not satisfy b) above) exhibiting subdiffusive behaviour in any dimension, and casted some doubts on the validity of the renormalization group analysis. Our result justifies this analysis for d > 2, independent transition probabilities and weak disorder. Actually, the proof itself is based on a related RG analysis. Whether diffusive behaviour can be shown to hold for d = 2 or for strong disorder is an open problem. Before giving the main ideas of the proof, we mention two related sets of results. The first ones concern symmetric random environments: for example, let us associate to each nearest neighbour bond (x, y) a random rate a(x, y). Consider a Markov process, with state space Zd and jumping from x to y at rate a(x, y). This is just a continuous time version of the model previously discussed. If we impose a symmetry a(x, y) = a(y, X) between the rate of jumping from x to y and from y to X, one can show, for all d and under mild assumptions on the distribution of a(x, y), that the walk is diffusive [9, 10, 111. Several variants of this model where the lattice is replaced by a continuum (Rd), have been studied [12]. Another related model concerns walks in a time-dependent random environment, also called direct polymers. Here “time ” is to be considered as one of the spatial dimensions. The walk moves in a prefered direction in that dimension. Therefore, at each “time”, the walk is in a different hyperplane perpendicular to that time direction and, thus, the environment changes at each time. For that model, Imbrie and Spencer [13] proved diffusive behaviour for weak coupling if the spatial dimension d > 2 (space-“time”, i.e. physical space here, d + 1 > 3). This result leads one to expect diffusive behaviour for d > 2 also for the time-independent random environment (our case). Indeed, there is evidently no difference between both environments as long as the walk does not revisit the same site. But since ordinary walks are transient in d > 2, the effect of several visits to the same site should be irrelevant. One easily sees that this argument is too crude: for the time dependent random environment considered in [13], the diffusion constant is independent of the strength of the disorder, which is not the case here (see (4)). Moreover, in the time dependent situation, one expects the walk to become superdiffusive at strong coupling, again unlike here.
J. Bricmont and A. Kupiainen I Rigorous renormalisation group and disordered systems
35
3. The Renormalisation Group trandormation We shall define our RG transformation and compute perturbatively how the RG flow goes. Our transformation is simply decimation in time, with a simple resealing (but no coarse graining) in space, in order to keep the distances covered by the walk of order one. More precisely, divide the time into intervals of length L2 : t = tJL2, t’ = 0, 1,2, . . . . Fix the positions of the walk at those times, x(O), x(L’), x(2L2) etc. . . and rescale them: x’(t’)
= L_‘x(t’L2)
)
(5)
where the resealing factor is chosen because we anticipate diffusive behaviour. Since x is in Zd, x’ is in (L-‘Z)? The probabilities of the paths x(t) are defined by (2,3), those of x’(t’) are also defined, simply by resumming over the intermediate values of x(t), t # t’L2. Clearly the induced process on x’(t’) is again Markovian, with transition probabilities: I-1 p’(x’, y’) = Ld
c
lg PM97 x(i + 1))
(6)
x(l)...*(L*-1)
with p(x, y) given by (2). The normalisation is really a probability density:
I
by Ld is chosen because p’(x’, y’)
dy’ Ax‘, Y’) = 1 >
where s dy ’ stands for the Riemann sum L -dS on (L -lZ)d. The scale L is fixed: the idea, as always in RG analysis, is to study the map (between Markov processes), defined by (6), for fixed L, and to use properties of the iterates of this map to study the long-term behaviour of the original process. Here the transformations form a semi-group: writing (6) in the form p’ = T~( p) we have that T: = rL” = p”. The reason for studying (6) is that one expects to be able to write p’(x, y) = 7% -Y) + b’(x, Y) and pn(x, y) = T”(x - y) + b”(x, y) where T”(x - y) converges towards the heat kernel T”(x - y)+exp(-(x - y)*/2D), with a renormalised diffusion constant D = D(E), while b”(x, y) is a random variable whose variance tends to zero: (b”(x, y)‘) = E2L-(d-21n .
(7)
36
.I. Bricmont and A. Kupiainen
I Rigorous renormalisation group and disordered systems
Let us see, heuristically,
why one expects (7) to hold and, therefore, why This is most easily done for a similar problem, but where space and time are continuous. There, (7) becomes a simple scaling argument: Let the probability density P(x, t) to be at x at time t, starting from 0, be the solution of the Fokker-Planck equation: aP/at =V2P - V- (bP), where b is a random force with covariance b,(x)b,( y) = .s2Saup 6(x - y). Then, the transformation (6) takes the form d = 2 is the critical dimension.
P’(x’, t’) = LdP(Lx’,
L2t’) .
P’ satisfies the equation
!$
=V2P’ - v* (b’P’)
where b’(x)’ = Z&Lx’) b;(x’)b;(y’)
)
has covariance
= ~*L~6,,6(Lx =
- Ly)
E2PdSa,S(X-
y) .
This calculation is similar to (7). The lattice calculation is somewhat more involved and will not be discussed here (see [2]). However, these variance calculations can only be used when the fields are small. They do not allow us to control the (rare) events where the fields are large. This is why a more detailed RG analysis is necessary: we treat separately the places where the walk may get trapped or become superdiffusive (locally) due to large b(x, y)‘s. With large probability, these events occur far away from each other. Moreover, using the fact that the variance runs down, one shows that the “effective” large fields, i.e. the b”(x, y) which are large, are even less probable than the original b(x, y)‘s. If, on all scales, the large fields are rare and thus far away from each other, they cannot “conspire” to prevent the walk from being diffusive. To summarize, our RG analysis is based on the following picture: wherever the field b(x, y) is small, diffusion takes place. This occurs, typically, over large distances and long times intervals, if E is small. Being diffusive, the walk “feels” a kind of averaged environment, which is still random but has a smaller variance. Thus, upon iteration, the randomness disappears. References [l] Ya.G. Sinai’, Th. Prob. Appl. 27 (1982) 256. [2] J. Bricmont and A. Kupiainen, in preparation.
J. Bricmont
and A. Kupiainen
[3] J. Bricmont and A. Kupiainen,
I Rigorous
renormalisation
group and disordered
systems
37
Commun. Math. Phys. 116 (1988) 539. D.S. Fisher, Phys. Rev. A 30 (1984) 960. B. Derrida and J.M. Luck, Phys. Rev. B 28 (1983) 7183. J.M. Luck, Nucl. Phys. B 225 (1983) 169. E. Marinari, G. Parisi, D. Ruelle and P. Windey, Commun. Math. Phys. 89 (1983) 1. R. Durrett, Commun. Math. Phys. 104 (1986) 87. V.V. Anshelevich, K.M. Khanin and Ya.G. Sinai, commun. Math. Phys. 85 (1982) 449. R. Kunnemann, Commun. Math. Phys. 90 (1983) 27. A. De Masi, P.A. Ferrari, S. Goldstein and W.D. Wick, J. Stat. Phys. 55 (1989) 787. G.C. Papanicolaou and S.R.S. Varadhan, in: Statistics and Probability: essays in honor of C.R. Rao; G. Killianpur, P.R. Krishaniah and J.K. Ghosh, eds. (North-Holland, Amsterdam, 1982) p. 547. [13] J. Imbrie and T. Spencer, Diffusion of directed polymers in a random environment. Preprint.
[4] [5] [6] [7] [8] [9] [lo] [ll] [12]
RIGOROUS
RENORMALISATION
GROUP AND DISORDERED
SYSTEMS
J. BRICMONT Znstitut de Physique Thkorique, Universitti 1348 Louvain-la-Neuve, Belgium
Catholique
de Louvain,
Chemin du Cyclotron,
2,
A. KUPIAINEN Mathematics
Department,
Rutgers
University,
New Brunswick,
NJ 08540, USA
We consider random walks in a non-symmetric random environment. We report a recent result, based on a Renormalisation Group approach, showing that for d > 2, these walks are diffusive when the disorder is weak.
1. Introduction
A standard way to model the effect of disorder and impurities in physical systems is to replace certain parameters of the system by random variables. Random walks in random environments provide a simple example of such a procedure. A random walk on Zd, with steps taken at unit times, is defined by a set of transition probabilities p(x, y), x, y E Zd, satisfying:
Pk
7
Y) 2
0
3
PC6Y) = 1 .
(1)
If the walk takes place in a homogeneous environment, these probabilities are translation invariant: p(x, y) = p(x - y). To model an inhomogeneous environment, one takes p(x, y) to be random variables themselves. The usual questions about random walks (recurrence, diffusive behaviour, . . .) are now to be answered for almost every set of transition probabilities. In this talk we shall sketch how rigorous Renormalisation Group methods can be used to prove that, above two dimensions, random walks in a “weakly” random environment diffuse just as ordinary random walks [2]. This is to be contrasted with the results of Sinai’ [l] who showed that a similar walk in one dimension moves, in a time t, only over a distance of order (log t)*. 0378-4371/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland)
32
J. Bricmont and A. Kupiainen
I Rigorous renormalisation group and disordered systems
Our RG analysis bears some resemblance with the one used before, but in a totally different context, to show that a small random field does not destroy the phase transition in the Ising model, above two dimensions [3]. However, in the Ising model, we dealt with equilibrium phenomena while here we deal with a non-equilibrium system. In either case, however, we do not, in order to apply the RG analysis, first reduce the model to a pure, non-disordered one, as is often done. Rather, we define in a some sense two RG flows, one on the statistical system itself (the random walks, or the Ising spins) and another one on the disorder (the environment or the random fields). We must emphasise that the fixed points towards which the RG flow converges are quite trivial, both for the Ising model and for the random walk. The extension of this method to non-trivial fixed points is an open problem.
2. Random walks in random environments Here we shall define more precisely the model, state our results and recall some previously known facts. We consider nearest-neighbour transition probabilities on Hd: (2) where if Ix-y]
= 1
otherwise defines the usual random walk and {b(x, y)]]x - y( = l} are random variables satisfying the following properties:
4
b) w-5 Y>, be’, Y') are independent 4 b(x, , _= .s2 (small), d) Prob
(
for x # x’ and identically distributed,
& + b(x, y) < eeN
In c) and below the bar denotes the average over the b-distribution. Condition a) means that the p(x, y) are transition probabilities, i.e. satisfy (1). Condition b) is in some sense what makes the problem difficult: there is no
1. Bricmont and A. Kupiainen
I Rigorous renormalisation group and disordered systems
33
relation between p( y, X) and p(x, y), i.e. between crossing a bond in one direction and in the opposite direction. This, in turn, may cause “trapping”: for example, we may have a nearest neighbour pair (w, z) with p(w, z) and p(z, w) almost equal to one and p(w, w’), w’ # z, p(z, z’), z’ # w almost zero. Then, once the walk reaches either w or z, it may be stuck in between these two sites for a long time. Since nothing forces p(w’, w), w’ # z or p(z’, z), z’ # w to be small, the walk can easily reach z or w. It would not be too bad if this was the only example. However, it is easy to see that this pattern occurs on all scales: even if each individual b(x, y) is small, they may be (un)suitably oriented in such a way as to “push” the walk and confine it, for a long time, in a given region. Of course, the walk cannot be trapped forever: this is insured by conditions d) and b): d) says that the probability that’some p(z, w) is almost one (which, by the sum rule (l), means that some p(z, z’) is small) is unlikely. Coming back to our first example, if p(z, z’) = emN, for all z’ # w, then the time it takes to escape from the trap is eN, which is not too large, ___ given the ~small density (=ePNis) of such events. Condition b) implies that b(x, y) = b(x, y’) and, by a), that b(x, y) = 0. So there is no net drift in the system. This, in turn, will imply that the large scale traps described above are even more unlikely than those involving only two sites. Finally, condition c) means that typical b(x, y)‘s are small. Under the above hypotheses, we show the following: define the mean square displacement as usual, t-1
(x2(t))=
c x2(t)lgoP(X(9?x(i + 1)) (x(i)):=1
(3)
with x(O) = 0. Then, if d > 2, lim t-e-
0
=
D(E)
t
with probability
one. Moreover,
II(&) = D(0) +
O(2).
(4)
‘So the behaviour is diffusive, but with a “renormalized” diffusion constant. One of the first results on random environments is due to Sinai [l] who considered the same model, but in one dimension. There, he proved, with no restriction on E, that (x’(t))
= (ln t)” .
34
J. Bricmont
and A. Kupiainen
I Rigorous renormalisation group and disordered
systems
So, it was natural to ask whether such a subdiffusive behaviour might also occur in higher dimensions. Using a renormalisation group calculation, Fisher [4] and Derrida and Luck [S, 61 concluded that, for d 2 2, normal, diffusive behaviour takes place. Strictly speaking, this calculation applies only to small E, but Fisher [4] argues that the same result should hold for all E. This conclusion is at variance with the numerical results of [7], which concern d = 2 and strong disorder (large E). Moreover, Durrett [8] constructed counterexamples (but with dependent transition probabilities, i.e. which did not satisfy b) above) exhibiting subdiffusive behaviour in any dimension, and casted some doubts on the validity of the renormalization group analysis. Our result justifies this analysis for d > 2, independent transition probabilities and weak disorder. Actually, the proof itself is based on a related RG analysis. Whether diffusive behaviour can be shown to hold for d = 2 or for strong disorder is an open problem. Before giving the main ideas of the proof, we mention two related sets of results. The first ones concern symmetric random environments: for example, let us associate to each nearest neighbour bond (x, y) a random rate a(x, y). Consider a Markov process, with state space Zd and jumping from x to y at rate a(x, y). This is just a continuous time version of the model previously discussed. If we impose a symmetry a(x, y) = a(y, X) between the rate of jumping from x to y and from y to X, one can show, for all d and under mild assumptions on the distribution of a(x, y), that the walk is diffusive [9, 10, 111. Several variants of this model where the lattice is replaced by a continuum (Rd), have been studied [12]. Another related model concerns walks in a time-dependent random environment, also called direct polymers. Here “time ” is to be considered as one of the spatial dimensions. The walk moves in a prefered direction in that dimension. Therefore, at each “time”, the walk is in a different hyperplane perpendicular to that time direction and, thus, the environment changes at each time. For that model, Imbrie and Spencer [13] proved diffusive behaviour for weak coupling if the spatial dimension d > 2 (space-“time”, i.e. physical space here, d + 1 > 3). This result leads one to expect diffusive behaviour for d > 2 also for the time-independent random environment (our case). Indeed, there is evidently no difference between both environments as long as the walk does not revisit the same site. But since ordinary walks are transient in d > 2, the effect of several visits to the same site should be irrelevant. One easily sees that this argument is too crude: for the time dependent random environment considered in [13], the diffusion constant is independent of the strength of the disorder, which is not the case here (see (4)). Moreover, in the time dependent situation, one expects the walk to become superdiffusive at strong coupling, again unlike here.
J. Bricmont and A. Kupiainen I Rigorous renormalisation group and disordered systems
35
3. The Renormalisation Group trandormation We shall define our RG transformation and compute perturbatively how the RG flow goes. Our transformation is simply decimation in time, with a simple resealing (but no coarse graining) in space, in order to keep the distances covered by the walk of order one. More precisely, divide the time into intervals of length L2 : t = tJL2, t’ = 0, 1,2, . . . . Fix the positions of the walk at those times, x(O), x(L’), x(2L2) etc. . . and rescale them: x’(t’)
= L_‘x(t’L2)
)
(5)
where the resealing factor is chosen because we anticipate diffusive behaviour. Since x is in Zd, x’ is in (L-‘Z)? The probabilities of the paths x(t) are defined by (2,3), those of x’(t’) are also defined, simply by resumming over the intermediate values of x(t), t # t’L2. Clearly the induced process on x’(t’) is again Markovian, with transition probabilities: I-1 p’(x’, y’) = Ld
c
lg PM97 x(i + 1))
(6)
x(l)...*(L*-1)
with p(x, y) given by (2). The normalisation is really a probability density:
I
by Ld is chosen because p’(x’, y’)
dy’ Ax‘, Y’) = 1 >
where s dy ’ stands for the Riemann sum L -dS on (L -lZ)d. The scale L is fixed: the idea, as always in RG analysis, is to study the map (between Markov processes), defined by (6), for fixed L, and to use properties of the iterates of this map to study the long-term behaviour of the original process. Here the transformations form a semi-group: writing (6) in the form p’ = T~( p) we have that T: = rL” = p”. The reason for studying (6) is that one expects to be able to write p’(x, y) = 7% -Y) + b’(x, Y) and pn(x, y) = T”(x - y) + b”(x, y) where T”(x - y) converges towards the heat kernel T”(x - y)+exp(-(x - y)*/2D), with a renormalised diffusion constant D = D(E), while b”(x, y) is a random variable whose variance tends to zero: (b”(x, y)‘) = E2L-(d-21n .
(7)
36
.I. Bricmont and A. Kupiainen
I Rigorous renormalisation group and disordered systems
Let us see, heuristically,
why one expects (7) to hold and, therefore, why This is most easily done for a similar problem, but where space and time are continuous. There, (7) becomes a simple scaling argument: Let the probability density P(x, t) to be at x at time t, starting from 0, be the solution of the Fokker-Planck equation: aP/at =V2P - V- (bP), where b is a random force with covariance b,(x)b,( y) = .s2Saup 6(x - y). Then, the transformation (6) takes the form d = 2 is the critical dimension.
P’(x’, t’) = LdP(Lx’,
L2t’) .
P’ satisfies the equation
!$
=V2P’ - v* (b’P’)
where b’(x)’ = Z&Lx’) b;(x’)b;(y’)
)
has covariance
= ~*L~6,,6(Lx =
- Ly)
E2PdSa,S(X-
y) .
This calculation is similar to (7). The lattice calculation is somewhat more involved and will not be discussed here (see [2]). However, these variance calculations can only be used when the fields are small. They do not allow us to control the (rare) events where the fields are large. This is why a more detailed RG analysis is necessary: we treat separately the places where the walk may get trapped or become superdiffusive (locally) due to large b(x, y)‘s. With large probability, these events occur far away from each other. Moreover, using the fact that the variance runs down, one shows that the “effective” large fields, i.e. the b”(x, y) which are large, are even less probable than the original b(x, y)‘s. If, on all scales, the large fields are rare and thus far away from each other, they cannot “conspire” to prevent the walk from being diffusive. To summarize, our RG analysis is based on the following picture: wherever the field b(x, y) is small, diffusion takes place. This occurs, typically, over large distances and long times intervals, if E is small. Being diffusive, the walk “feels” a kind of averaged environment, which is still random but has a smaller variance. Thus, upon iteration, the randomness disappears. References [l] Ya.G. Sinai’, Th. Prob. Appl. 27 (1982) 256. [2] J. Bricmont and A. Kupiainen, in preparation.
J. Bricmont
and A. Kupiainen
[3] J. Bricmont and A. Kupiainen,
I Rigorous
renormalisation
group and disordered
systems
37
Commun. Math. Phys. 116 (1988) 539. D.S. Fisher, Phys. Rev. A 30 (1984) 960. B. Derrida and J.M. Luck, Phys. Rev. B 28 (1983) 7183. J.M. Luck, Nucl. Phys. B 225 (1983) 169. E. Marinari, G. Parisi, D. Ruelle and P. Windey, Commun. Math. Phys. 89 (1983) 1. R. Durrett, Commun. Math. Phys. 104 (1986) 87. V.V. Anshelevich, K.M. Khanin and Ya.G. Sinai, commun. Math. Phys. 85 (1982) 449. R. Kunnemann, Commun. Math. Phys. 90 (1983) 27. A. De Masi, P.A. Ferrari, S. Goldstein and W.D. Wick, J. Stat. Phys. 55 (1989) 787. G.C. Papanicolaou and S.R.S. Varadhan, in: Statistics and Probability: essays in honor of C.R. Rao; G. Killianpur, P.R. Krishaniah and J.K. Ghosh, eds. (North-Holland, Amsterdam, 1982) p. 547. [13] J. Imbrie and T. Spencer, Diffusion of directed polymers in a random environment. Preprint.
[4] [5] [6] [7] [8] [9] [lo] [ll] [12]