On the extinction of Dickman's reaction- diffusion processes

Physica A 186 <1992 578-590 North-Holland

On the extinction of Dickman's reactiondiffusion processes Makoto

K a t o r i a'' a n d N o r i o K o n n o b

aDepartment of Physics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan bDepartment of General Education, Muroran Institute of Technology, Mizumoto-cho, Muroran 050, Hokkaido, Japan Received 1 November 1991 Some comments on the extinction of processes are given for the reaction-diffusion processes recently proposed by Dickman as mathematical models for chemical reactions on catalytic surfaces. A brief review of mean-field-type approximations (MFA) is presented for three models; the single annihilation model (SAM), the pair annihilation model (PAM) and the triplet annihilation model (TAM). Two theorems on the extinction are proved. The former supports the MFA predictions for the SAM. The latter gives a qualitative correction to the phase diagram obtained by the MFA for the PAM in low dimensions (d ~<2). In order to obtain the latter theorem, we discuss the relationship between the PAM and the branching annihilating random walk of Bramson and Gray.

I. Introduction R e c e n t l y D i c k m a n [1, 2] p r o p o s e d s o m e r e a c t i o n - d i f f u s i o n p r o c e s s e s as m a t h e m a t i c a l m o d e l s for c h e m i c a l r e a c t i o n s on c a t a l y t i c surfaces. His m o d e l s c o n s i s t o f t h e f o l l o w i n g t h r e e p r o c e s s e s . (i) A d s o r p t i o n o f a t o m s o n t h e s u r f a c e d u e to a t t r a c t i v e i n t e r a c t i o n s is r e p r e s e n t e d by c r e a t i o n of p a r t i c l e s o n t h e s u r f a c e b y c o n t a c t p r o c e s s e s . (ii) N e i g h b o r i n g p a r t i c l e s o n t h e surface r e a c t to m a k e a c o m p o u n d m o l e c u l e , w h i c h d e s o r b s with a c o n s t a n t r a t e l e a v i n g e m p t y sites o n t h e surfaces. T h i s p r o c e s s is r e p r e s e n t e d by m u l t i - p a r t i c l e a n n i h i l a t i o n . (iii) J u m p i n g p r o c e s s e s o f p a r t i c l e s a r e also i n t r o d u c e d to r e p r e s e n t diffusion d u e to t h e r m a l fluctuations. D i c k m a n g a v e s o m e s o l u t i o n s o f m e a n - f i e l d - t y p e a p p r o x i m a t i o n s a n d p e r f o r m e d c o m p u t e r s i m u l a t i o n s to s h o w t h e e x i s t e n c e o f p h a s e t r a n s i t i o n s f r o m an e x t i n c t i o n s t a t e ( d e v o i d o f particles) to an active s t e a d y s t a t e (with p a r t i c l e s surviving e v e n in t h e limit t - - - ~ ) [1,2]. It is Present address: Department of Physics, Faculty of Science and Engineering, Chuo University, Bunkyo-ku, Tokyo 112, Japan. 0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved

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important to determine phase diagrams for these models for testing the validity of these models to describe the poisoning of catalysts. Especially competition between diffusion and multi-particle annihilation should be studied. From the viewpoint of statistical physics and the study of interacting particle systems, Dickman's models will attract much attention, because they are related to important interacting particle systems which have been well studied by mathematicians [3-5]. They can be viewed as the combinations of the exclusion processes and some modified versions of the contact process (CP) of Harris [6]. The active (survival) states of his models may be related to the nontrivial upper invariant measure of the CP, which does not satisfy the reversibility (i.e. the condition of the detailed balance). The multi-particle annihilation makes the processes fail to have a certain monotonicity property known as attractiveness. However the pair annihilation model of Dickman is closely related to the branching annihilating random walk (BARW) studied by Bramson and Gray [7]. In this paper, we present some comments on the region of the extinction phase in phase diagrams for some of the Dickman models. In this phase the process dies out for any initial state. We discuss the conditions of the extinction of processes for the single annihilation model (SAM) and the pair annihilation model (PAM). In section 2, formal generators of the SAM, the PAM and the triplet annihilation model (TAM) are given and a brief review of the meanfield-type approximations is presented. Section 3 is devoted to the SAM, where the duality of this process and the submodularity of the survival probability are used to derive the conditions of extinction. In section 4, we explain the relation between the PAM and the B A R W and give a statement on the extinction region in low dimensions (d~<2) following the argument of Bramson and Gray. Future problems are given in section 5.

2. Models and mean-field-type approximations 2.1. The Dickman models [1, 2] The reaction-diffusion models of Dickman are processes whose state space is X = {0, 1}z~: at each takes the value 0 or 1, representing a vacant site respectively. The formal generator of the processes = ~'2cre + ~ a n + ~'~dif •

continuous time Markov site x E 7/d a variable ~(x) and a particle on a site, O consists of three parts: (2.1)

Let C(X) be a set of the continuous functions on X and let Y be a collection of all finite subsets of 2rd. We define 7/A E X for A E Y as

M. Katori, N. Konno / On the extinction of Dickman's processes

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rta(u) =

1 - 7/(u),

if u E A ,

rt(u),

otherwise.

The first term of eq. (2.1), O .... denotes the creation process with a rate A by a nearest-neighbor interaction (the contact process); for f ~ C(X), [2cr~f(r/) = A Z

Z

[1 - r/(x)] ~7(Y) [f(r/{~}) - f ( r / ) ] .

(2.2)

x~Z d y:ly-xl-I

The diffusion term ~'~dif is the exclusion process with jumping to neighboring sites at a rate D ; for f ~ C(X), ~diff('r/) = D ~]

~]

rt(x)[1 - ~l(y)][f(*l {x'y}) - f ( r / ) ] .

(2.3)

x~Z d y:ty-x[=l

For each model the annihilation t e r m

~'~an

is given as follows for f ~ C(X).

Single annihilation model (SAM), L2a,f(~/) = ~] ~/(x)[f(~/{x}) _f(~/)] .

(2.4a)

xEZ d

Pair annihilation model ( PAM), aanf('l~)

d = Z £ •(X) g~(X + e i ) [ f ( ~ {x'x+ei}) - f('lT) ] . xE~d i= 1

(2.4b)

Triplet annihilation model (TAM), d [~anf(r/)

= Z

Z

X@7/d i 1

rl(X) '0( X + ei) 17(x + 2ei) [f(rl {x'x+ei'x+zei})

-- f(r/)]

.

(2.4c)

Here the ei's are the unit vectors of 7/a. There may be a unique Markov process rl, on X corresponding to the Markov semigroup S(t) generated by g2 for each model [3].

Remark 2.1. The definition of parameters describing the rates of the processes given here is slightly different from the original one by Dickman [1]. We choose the one above for the convenience of showing the relation with other interacting particle systems.

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2.2. Mean-field-type approximations [1] We assume the uniqueness of the upper invariant measure/2 ---limt~=61S(t), where 61 is a point mass on rt =- 1. We describe the expectation of a function f @ C ( X ) on the measure as Ea[f(7/)]. Thus /2 is the measure where the stationary condition, E a [Of(n) ] = 0 ,

(2.5)

is satisfied for any f E C(X). In the present paper, mean-field-type approximations (MFA) mean the procedures to approximate 12 by assuming simple forms. We introduce two kinds of MFA. Let g ( - ) be an appropriate simple function, e.g. g(a) = a or g(a) = 1 - a.

One-point-level M F A . Let ao(X ) be a [0, 1]-valued function and the measure/~ is assumed to be decoupled as

E~[ 1-[ g07(x))] = 1-[ ao(X) , xEA

x~A

VA E Y.

(2.6)

A n d a o is determined as a function of h and D to satisfy eq. (2.5) for f(rl) = g(rl(x)), V{x} ~ 'r.

Pair level M F A . Let a11(x ) and alz(x, y) be [0, 1]-valued functions and the measure is assumed to be decoupled to the following form: E a [ I-[ g(7/(x))] = I~ a l , ( x ) × xEA

xEA

I-[

a12(x, y),

VA E Y.

{x,y}CA:lx-yl= 1

(2.7) A n d all and O~12are determined as functions of A and D to satisfy eq. (2.5) for both f07) = gOT(x)), V{x} E Y and f(r/) = gOT(x)) g(rl(y)), V{x, y} E Y with Ix - y l = 1.

Because the upper invariant measure/~ is translation invariant, the functions a do not depend on the choice of sites. Thus we can determine these approximate measures by simple calculations. In the present MFA, the extinction of the process is expected when only a solution a ~- 0 is allowed, while the survival states are described by nonzero solutions.

Remark 2.2. For some interacting particle systems, it was proved that one-

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M. Katori, N. Konno / On the extinction o f Dickman's processes

level M F A with the product measure becomes exact in the hydrodynamic limit (roughly speaking, in the limit D---~ ~) [8]. The pair-level approximation may be considered as a better approximation for the case D < ~ with corrections to such a simple product measure. Now we show the solution of the M F A for each model. For this model we take g ( a ) = 1 - a to make a MFA. (The reason for this choice will be clarified in the next section.) In the one-level M F A , only a solution s 0-=0 is obtained for A < 1 / 2 d . When A > 1 / 2 d , another solution, ~o = ( 2 d A - 1)/2dA, is also obtained. The pair level M F A enlarges the predicted region of the extinction phase in the phase diagram as D < [1 - (2d - 1 ) a ] / [ ( 2 d - 1)(2dA - 1)] as shown in fig. la. SAM.

In this case we make the simplest choice; g ( a ) = a. The one-level M F A predicts no extinction region. However, we can find the region in the pair-level M F A ; the unique solution is %1 = a~2 ~-0 for D <~ 1 / [ 2 ( 2 d - 1 ) ] - A (see fig. lb). PAM.

TAM. Again g ( a ) = a. As in the PAM, we have to consider the pair-level M F A to obtain the extinction region. The predicted phase diagram is shown in fig. lc for the one-dimensional case, where the phase boundary is given by (2A 3 - 3A 2 - 9A - 9AD) - (2A 2 - 9A - 9 D ) V ' - ~ + 6A = 0. 2 . 3 . As discussed by Dickman in ref. [1], the diffusion process effectively inhibits the multi-particle annihilation processes and thus it is expected that the extinction region becomes smaller as the rate of diffusion D becomes larger. For the PAM and the T A M , the M F A predicts the existence of Remark

=

;o 0,,0

0.5

1.0 a

1~

X

0,0

O,Oo 0,5 b

1~0

X

0.00

0.06

0,,10 0.15 0,20 c

X

Fig. 1. Phase diagrams obtained by the MFA are given for (a) SAM, (b) PAM and (c) TAM. Here we show the one-dimensional case and hatch the region where the processes become extinct. It is expected following the MFA that the processes can survive for any h >0 if D > D* in the PAM and the TAM.

M. Katori, N. Konno I On the extinction o f Dickman's processes

583

an upper limit of D in the extinction phase in the phase diagram in the A-D plane [1], which is denoted by D* in figs. lb and c.

3. On the extinction region of the SAM The SAM consists of the CP and the exclusion process. Because both processes are self-dual with their coalescing dual processes [3], so is the as shown below. Let A, = {x: ~?,(x) = 1}, then A t is a Markov process on is a coalescing dual of "Or It is easy to find the following duality for any and A E Y, PO[7/t(x) = 0, Vx E A]

=

PA[rI(x )

=

0,

VX E A , ] .

of the SAM, Y and ~7E X

(3.1)

Consider the case that the initial state is 61 (i.e. ~/= 1) and take the limit t---~ in eq. (3.1). Then we obtain 1 - Ea{ l~ [ 1 - r/(x)]} = (r(A).

(3.2)

xEA

Here tr(A) is a survival probability of the process with the initial state {x: ~?o(X) = l} = A;

o-(A) = lim pA[A t ~ 0].

(3.3)

By using the survival probability, the extinction of the process is characterized as

the process is extinct<=> o-(A)= 0,

VA E Y.

(3.4)

We define H(rh A) =- I]xE A [1 - rt(x)]. Because H(r/, A) + H0?, B) ~< H(-q, A U B ) + H ( r / , A A B ) for any A, B E Y , we obtain the following inequality (submodularity [3]) by using the relation (3.2): cr(A U B) + o'(A n B) ~< or(A) + o'(B).

(3.5)

By this inequality, the extinction of the process (3.4) will follow once we have shown that (r({x})= 0 for any singleton {x}. That is, it is sufficient to show that E~z[~7(x)] = 0 to conclude the extinction of the SAM. Now we give a theorem on the extinction of process for the SAM.

Theorem 3.1. In the parameter region where the M F A predicts the extinction, the process of the SAM indeed dies out.

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Proof of theorem 3.1. We use the following notation: for A E Y p(Z)-~ E~( I-[ ~7(x)).

(3.6)

xEA

Let A l = { x }, A2 = { x , x + el}, A3 = { x , x + e l , x + 2el}, A 4 = { x , x + 2el}, A 5 = {x, x + e 1, x + e2} and A 6 = {X + e l , X -t- e 2 } . For f(rl) = r/(x) and f07) = ~7(x)~7(x + el), eq. (2.5) gives the following equalities: (3.7a)

(2dA - 1)p(A1) - 2dAp(A2) = O, AR(A1)

--

[A + 1 + (2d - 1)D]p(A2) + A[p(A4) - p(A3)I

+ 2A(d - 1)[p(A6) - p(As) ] + Dp(A4) + 2(d - 1)Dp(A6) = O. (3.7b) Because p(Ai) >! 0, eq. (3.7a) implies that p(A1) = P(A2) = 0 for A < 1/2d. Next we eliminate p(A2) in eq. (3.7b) by using eq. (3.7a) and obtain

{ [ ( 2 d - 1)A- 11 + ( 2 d - 1)(2dA= 2dA2[p(A4) -

p(A3) ] +

1)D}p(A,)

4d(d -

1)A2[p(A6) - p(As) ] +

2dADp(A4)

(3.8)

+ 4d(d - 1)Op(A6).

Noticing eqs. (3.2) and (3.6), we find that the correlation inequalities, P ( A 4 ) - P(A3)i>0 and p ( A 6 ) - p ( A s ) / > 0 , are obtained from the inequality (3.5). Then eq. (3.8) implies (3.9)

{ [ ( 2 d - 1 ) A - 11 + ( 2 d - 1 ) ( 2 d A - 1 ) D } p ( A , ) > / 0 . Therefore, p(A,) =0.

if

[ ( 2 d - 1 ) A - 1] + ( 2 d - 1 ) ( 2 d A - 1 ) D < 0 ,

it

follows

that

Remark 3.1. The same argument has been used to show ergodicity for the basic CP in refs. [9, 10]. For the basic CP, see also ref. [11].

4. On the extinction region of the PAM in low dimensions

In the PAM, the simple duality (3.1) does not hold. Then we have to analyze the original process ~t to know the condition for the extinction of the process. It should be noticed that the present process is not attractive. Here an attractive

M. Katori, N. Konno / On the extinction o f Dickman's processes

585

spin system is one in which the addition of an extra occupied site to the state of the process does not decrease the rate at which any of the vacant site becomes occupied, nor does it increase the rate at which the other occupied sites are vacated. In the pair annihilation model, the addition of a new particle at a vacant site neighboring particles may reduce rather than increase the total number of particles. Bramson and Gray [7] introduced a Markov process which they called the branching annihilation random walk (BARW). In this process, a particle gives birth to a new particle at one of the neighbouring sites with a rate 1/2d (branching) and a particle jumps to one of the neighboring sites with a rate p/2d (random walk). And if a particle lands on a site which is already occupied, either by jumping there or as the result of branching, then both particles disappear (are annihilated). This process is closely related to the PAM and we find the argument which was given for the B A R W is also applicable to the present problem. In order to clarify the relationship between these two processes, we here define a generalized process which includes both of them as special cases.

Definition 4. i. A generalized branching annihilating random walk with parameters (/3~, /32; /33, /34) ( G B A R W (/31, /32; /33, /34)) is the Markov process generated by the following generator: for f E C(X),

O~,.~.~3,~,f(n) = ~

~

"o(x){/31[1-'o(y)l +/33"q(Y)}

xEZ d y:ly--x[=l

X [ f('r/{y}) -- f('o)] + Z

E

xEZ a y:]y-x]=l

n(X){/3211- 7/(y)]

+/34~l(Y)}[f(71 {x'y}) - f(n)l •

(4.1)

This process consists of four processes, which are parametrized by the rates/31, /32, /33 and/34, respectively. The processes with the rates/33 and /34 a r e single and pair annihilation processes, respectively.

Remark 4.1. It is easy to find that the B A R W of Bramson and Gray is the G B A R W (1/2d, p/2d; 1/2d, p/2d) and that we obtain the PAM by letting /31 = A,/32 = D,/33 = 0 and/34 = 1 in the GBARW. When we consider problems which are independent of the time-scale (e.g. the problem whether the process is extinct or not in the limit t--> ~), the same answer may be obtained both for G B A R W (A, D ; 0 ; ½) and G B A R W (1, D/A;0, 1/2A).

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M. Katori, N. Konno / On the extinction of Dickman's processes

R e m a r k 4.2. In ref. [7] B r a m s o n and G r a y proved that starting from a finite

n u m b e r of particles the B A R W becomes extinct with probability 1 if the jumping rate is sufficiently high; p >> 1, when d ~< 2. As m e n t i o n e d in r e m a r k 4.1, it is expected that the B A R W with high p may c o r r e s p o n d to the PAM with low A if D is finite, when we discuss the stationary state. Actually we find that the argument of Bramson and G r a y can be applied to the present p r o b l e m and obtain the following result. T h e o r e m 4.1. W h e n d ~< 2, the PAM with finite D becomes extinct for any initial state with finite particles, if A is sufficiently small.

In the proof, we use the recurrence of a modified version of r a n d o m walks. Thus this t h e o r e m is valid only for d ~< 2. We will give a sketch of the proof in the appendix. As shown in section 2, the pair level M F A suggests for the PAM that the value of D of the phase boundary between the extinction region and the survival region increases monotonically as A decreases and terminates at a finite D* = 1 / [ 2 ( 2 d - 1)], and that the process can survive for any A > 0 if D > D* (see fig. lb). T h e o r e m 4.1 implies that in low dimensions (d~<2), the true phase b o u n d a r y is qualitatively different from this M F A prediction. That is, as illustrated in fig. 2, we will find a narrow but finite region of extinction for any finite D. In other words, D* = ~ for d ~< 2. We expect that D* becomes finite for higher dimensions d >~ d 0 with some d o (which may be 3, because the r a n d o m walk is transient for d ~> 3).

t

D~=oo

k Fig. 2. An illustration of the phase diagram of the PAM in low dimensions (d ~<2). Contrary to the MFA prediction (see fig. lb), we will find a narrow but finite region of the extinction for any D < :~ (as shown by a hatched region).

M. Katori, N. Konno / On the extinction o f Dickman's processes

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5. Summary and future problems In the present paper, we gave some comments on the extinction of the reaction-diffusion processes proposed by Dickman [1, 2]. These processes are simple mathematical models of chemical reactions on catalytic surfaces, which consist of three processes: creation of particles by contact process with a rate A, (multi-)particle annihilation with a rate 1, and a jumping process with a rate D. One expects transitions from the surviving active phase to the extinction phase in some parameter region in the A-D plane. At first we showed the solutions of the MFA for three models, the SAM, PAM, and TAM [1]. The phase diagrams obtained by the approximations are given in fig. 1. Then we gave two theorems on the extinction. The first one (theorem 3.1) for the SAM supports the MFA predictions. On the other hand, the second theorem (theorem 4.1) implies that it is necessary to give qualitative correction to the MFA for the PAM when the dimensionality d ~<2. We introduced the GBARW to discuss the relationship between the PAM and the BARW of Bramson and Gray [7]. In addition to the extinction theorem mentioned in remark 4.2, Bramson and Gray proved that the BARW survives with positive probability if the jumping rate p is low enough relative to the branching rate [7]. Thus it is expected that the PAM will also survive for sufficient large A and that the phase boundary between the extinction phase and the survival phase actually exists at some intermediate values of A. Recently we have obtained the following theorem on the survival of the one-dimensional PAM [12]. T h e o r e m 5 . 1 . The one-dimensional PAM survives with positive probability for any initial state, if A is sufficiently large.

By theorems 4.1 and 5.1, it is confirmed that there occurs phase transition from the extinction phase to the survival phase in the one-dimensional PAM in stationary states for all D < ~. The reason why we can find a phase transition even in one-dimensional system is because the survival state of the PAM may not be described by the Gibbs state with finite-range interactions. This is caused by the lack of the detailed balance in this state. Both from the view points of physics and mathematics, the behavior of the phase boundary of the TAM is very interesting. As shown by the MFA, the essential competition between the multi-particle annihilation and the diffusion is expected in the intermediate region of the phase diagram. It would be important to clarify the condition of the reentrant phase transitions, which were observed in computer simulations [1,2], in the interacting particle systems without the property of attractiveness.

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M. Katori, N. Konno / On the extinction of Dickman's processes

Acknowledgement The present authors would like to thank R. Dickman for sending his paper previous to publication. They are also grateful to Prof. K. Uchiyama for showing the basic idea of theorem 4.1 and leading their attention to the paper by Bramson and Gray. This work is partially financed by the Grant-in-Aid for Encouragement of Young Scientists of the Ministry of Education, Science and Culture (Japan).

Appendix. A sketch of the proof of theorem 4.1 Let ~t be the G B A R W (1, D/A; 0, 1/21). Following Bramson and Gray [7], we introduce the processes ~xt and c~, associated with ~, when G0(x) = 1 as ~:~ = those particles in ~t which are descended from the particle at x at time 0, C X ~t = those particles in sc, which are descended from particles not at x at time 0. Of course, ~ U c ~ = ~,. To prove theorem 4.1, it is enough for us to prove the following proposition. Proposition A . 1. If d ~<2, for sufficiently small A and for x s.t. (?0(x) = 1,

E e [ I ~ [ ] ~< a < 1.

(A.1)

Theorem 4.1 follows from proposition (A.1), since summing eq. (A.1) over {xl ~0(x) = 1} implies that E~[[ ~11] ~< 1 01, and therefore, by induction, that Ee[l l] nl 01. This quantity becomes zero as n---~% which implies the theorem 4.1. This proposition can be proved just following the estimations which were performed by Bramson and Gray to prove their proposition 2 in ref. [7]. Thus, here we give a sketch of the proof and only explain some necessary modifications. A sketch o f the p r o o f o f proposition A . 1. We first introduce the following

notations. Let ~-= time at which pair annihilation between a particle from ~ and a particle from c~ first occurs, 0-1 = time at which sc~ first branches, and let or = min{r, o-1}. Then we define

M. Katori, N. K o n n o / On the extinction o f D i c k m a n ' s processes

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0.2 = time at which ~:~ branches again (= o~ if 0. = z). Let A 1 = event that 0. ~< 1 - e, A 2 = event that 0-2 > 0- + e, A 3 = event that r ~ (0-1, 0.1 + e], where 0 < e ~< 1. It is easy to find that

Ee[I ~'~11= E~[I ({I; +

A~] + E'~[I s¢11; A , O A~]

Ee[I ~'{I; A ,

N A 2 N A3] +

E'[I~I;

A, N A 2 N Ad.

(A.2)

By an estimation similar to that by Bramson and Gray, we find that each of the first three quantities on the right-hand side of eq. (A.2) becomes less than ¼for sufficiently small e and small h. In order to estimate the fourth quantity, we have to modify the argument of Bramson and Gray. Either ~ = 0 or I scx[ = 2. Because we want to estimate the u p p e r bound, h e r e a f t e r we consider the latter case (i.e. 0. = 0.~). In this case we define K = time at which pair annihilation between particles both from ~ first Occurs. By definition K > o - 1. A n d if K < 0 . 1 + e , it follows that I ~ , + ~ t - - o under A 2 O A 3. On the other hand, if K > 0.~ + e, I~:X+el = 2 under A 2 O A 3 and we find

E [I

A , n A 2 n A3] ~<2P[K > 0., + e l .

(A.3)

By the strong M a r k o v property, ~ l + , + , with t 1> 0 is pathwise dominated by the binary branching process Z, having the initial state [ ~x~ +~ [ with a branching rate 1. T h e r e f o r e

Ee[I =11; A 1 n

A 2 n A3] ~ E[Z,_(o. +e); A , N A 2 n A3]

<~E[Z1; A 1 n A 2 n As] x

=eE [I~,+~[;A~AA:AA3].

(A.4)

C o m b i n i n g eqs. (A.3) and (A.4) gives A , n A 2 n A3] <~2ee[K > 0.1 + e].

(A.5)

N o w we introduce a modified version of a r a n d o m walk which we call a (Pl, P2)-random walk.

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M. Katori, N. Konno / On the extinction of Dickman's processes

D e f i n i t i o n A . 1. A ( P l , Pc) - r a n d ° m w a l k is a p r o c e s s w h e r e e a c h p a r t i c l e j u m p s to o n e o f t h e n e i g h b o r i n g sites with a r a t e Pt if the site is e m p t y a n d with a r a t e P2 if t h e site is a l r e a d y o c c u p i e d by o t h e r p a r t i c l e s . It is n o t difficult to s h o w t h a t P[K > orI + e] ~< P [ t w o ( 1 , 1 / D ) - r a n d o m w a l k ers i n i t i a l l y at n e i g h b o r i n g sites d o n o t m e e t by t i m e eD/A]. T h e f o l l o w i n g lemma may be proved. L e m m a A . 1 . If O < p , < ~ and O
t h e ( p , , p 2 ) - r a n d o m w a l k is re-

T h i s l e m m a i m p l i e s t h a t for d ~< 2 t h e r i g h t - h a n d side of ( A . 5 ) goes to z e r o as A ~ 0 w h e n D is finite. T h u s for e a c h finite D we m a y find such a r e g i o n with sufficiently s m a l l A as E q [ ~ l l ; A 1 n a 2 n A 3 ] < 1.

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