Jun 1, 1991 - To cite this article: C. Aslangul et al 1991 EPL 15 251. View the article ... Laboratoire de Physique Statistique, Collbge de France. 3, rue d'Ulm, ...
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1 June 1991
EUROPHYSICS LETTERS
Europhys. Lett., 15 (3), pp. 251-254 (1991)
Random Walks on Biased Bethe Lattices. C. ASLANGUL(*),M. BARTHELEMY (*), N. POTTIER (*) and D. SAINTJAMES(**) ('1 (*) Croupe de Physique des Solides(s9),Tour 23, Universitb Paris V I I 2, place Jussieu, 75251 Paris Cedex 05, France (**) Laboratoire de Physique Statistique, Collbge de France 3, rue d'Ulm, 75231 Paris Cedex 05, France (received 4 February 1991; accepted in final form 2 April 1991) PACS. 05.40 - Fluctuation phenomena, random processes and Brownian motion.
Abstract. - The random walk on a biased ordered Bethe lattice is studied by elementary methods for the continuous time model. The average coordinate and the dispersion are obtained as a function of the bias. It is shown that the behaviour is strongly dependent on the polarization, according to whether the latter overcompensates or not the number of links leading to a higher shell. For the undercompensation a conventional diffusion is observed, while for the overcompensation a kind of localization takes place. For the exact compensation the motion shows a critical slowing-down.
The study of random walks on lattices has been the subject of renewed interest [l-31. In particular the case of Bethe lattices has been recently investigated by Cassi for discrete time hoppings of the walker [2,3]. We present here an alternate derivation for continuous time motions as described by conventional master equations, and we obtain the average coordinate and dispersion. A Bethe lattice is characterized by its coordination number 2. It may be divided into shells. The first one contains the Z neighbour sites of the origin 0, the second the Z(Z - 1) neighbours of the first shell sites, and so on. Thus the n-th shell contains Z(Z - l),-' sites. We shall consider an x,, the above reasoning would lead to a negative current which is unphysical. This shows that a given walker will certainly come back to the origin, and then restart from it, etc. In other words the probability of return to the origin is equal to 1, a result already obtained by Cassi [3]. Note that this does not imply that the average position oscillates, since two different walkers are not in x, we shall very simply obtain the asymptotic behaviour by remarking that, since the P's tends towards constants, their derivatives vanish and so does dWdt. One readily obtains @(Y, m,
since Po(")= XP,(CQ).
(1- x) exp [ip] = (Z - x)exp [ip]
+ x exp [ - ip] - 1 Po(CQ), + x exp [- ip] - z
254
EUROPHYSICS LETTERS
Combining eq. (13) with the relation between @(O, t ) and Po(t)imposed by the norm conservation, one deduces
Po+ (2%- Z)/2x ,
for t +
.
(14)
This result is in agreement with the value obtained by Cassi who predicts P,,(W)= 0 when the number of jumps is odd, and twice the value (14) when this number is even. By taking the proper derivatives at p = 0 , one obtains
These results may be checked for the special case 2 = 2, i.e. y = 0. In this case the walker’s motion is confined within the origin and the first shell. The two master equations relating Po and P I are easily solved and yield 1 (1 + exp [- 2ZwtI) ; Po(t)= -
1 Pl(t)= 2 (1 - exp [- 2ZwtI). 2 From these expressions one immediately derives the values predicted by (15). One can also show that the conventional Laplace transform P,(x) of the probability to be at the origin is, for any value of x, given by
PO(d =
2(2 - x)
+ V(X+ ZW)’- 4x(Z - x ) d
(2 - ~ x ) ( + x ZW) 2
(16)
This expression is in agreement with the result of Cassi. Clearly when the proper limit is taken for x + O , one recovers the various forms given above. Moreover one can easily show, by using (16), that at the critical point x = x,,Po(t)is given by Po(t)= exp [- Zwt]lo(Zwt), (17)
so that P&) behaves like liv-at large times as predicted by Cassi [31. This behaviour is reminiscent of a critical slowing-down. In conclusion, we have shown by elementary calculations that the motion of a walker on a biased Bethe lattice presents two different kinds of behaviour which can be easily understood. Indeed the Brownian motion of the walker is strongly dependent on the configuration, i . e . on the choice between the various links on a given site. For a biased lattice, the greater number of links leading to a higher shell may be overcompensated by the polarization. When this occurs the tendence to go back to the origin leads t o a kind of localization of the walker, while in the other case a conventional diffusion process is observed. When the compensation is exact, a kind of slowing-down takes place. This leads to consider the passage from diffusion to localization as akin to a phase transition, where the role of the order parameter is played, e . g . , by Po and that of temperature by the parameter x. REFERENCES [l] HUGHESB. D. and SAHIMI M., J . Stat. Phys., 29 (1982) 781. 121 CASSI D.,Europhys. Lett., 9 (1989)627. [3] CASSI D.,Europhys. Lett., 13 (1990)583.
