We consider the motion of a discrete random surface interacting by exclusion
with a random wall. The heights of the wall at the sites of $\Z^d$ are i.i.d.\
random variables. Fixed the wall configuration, the dynamics is given by the
serial harness process which is not allowed to go below the wall. We study the
effect of the distribution of the wall heights on the repulsion speed.Comment: 8 page

Fontes, L. R. G.

Vachkovskaia, M.

Yambartsev, A.

English

arXiv

Numérisation de Documents Anciens Mathématiques

Ann. I. H. Poincaré – PR 42 (2006) 207–213www.elsevier.com/locate/anihpbRepulsion of an evolving surface on walls with random heightsL.R.G. Fontes a,∗,1, M. Vachkovskaia b,2, A. Yambartsev a,3a Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo,rua do Matão 1010, CEP 05508-090, São Paulo SP, Brazilb Department of Statistics, Institute of Mathematics, Statistics and Scientific Computation, University of Campinas,Caixa Postal 6065, CEP 13083-970, Campinas SP, BrazilReceived 22 April 2004; received in revised form 18 January 2005; accepted 18 March 2005Available online 21 June 2005AbstractWe consider the motion of a discrete random surface interacting by exclusion with a random wall. The heights of the wallat the sites of Zd are i.i.d. random variables. Fixed the wall configuration, the dynamics is given by the serial harness processwhich is not allowed to go below the wall. We study the effect of the distribution of the wall heights on the repulsion speed. 2005 Elsevier SAS. All rights reserved.RésuméOn considère le mouvement d’une surface aléatoire discrète interagissant par exclusion avec une paroi aléatoire. Les hauteursde la paroi sont des variables aléatoires i.i.d. Une fois fixée la configuration de la paroi, la dynamique est donnée par un processusde harnais qui ne peut descendre en dessous de la paroi. On étudie l’effet de la distribution hauteurs sur la vitesse de répulsion. 2005 Elsevier SAS. All rights reserved.MSC: 60K35; 82C41Keywords: Harness process; Surface dynamics; Entropic repulsion; Random environment* Corresponding author.E-mail addresses: lrenato@ime.usp.br (L.R.G. Fontes), marinav@ime.unicamp.br (M. Vachkovskaia), yambar@ime.usp.br(A. Yambartsev).1 Partially supported by CNPq grants 300576/92-7 and 662177/96-7 (PRONEX) and FAPESP grant 99/11962-9.2 Partially supported FAPESP grant 99/11962-9 and CNPq grant 306029/2003-0.3 Supported by FAPESP grant 02/01501-9.0246-0203/$ – see front matter  2005 Elsevier SAS. All rights reserved.doi:10.1016/j.anihpb.2005.03.003208 L.R.G. Fontes et al. / Ann. I. H. Poincaré – PR 42 (2006) 207–2131. IntroductionThis paper is part of a project aiming to understand the effect of the interaction with walls on the evolution of ad-dimensional random surface in (d + 1)-space.The evolving random surface is modeled by the harness process introduced by Hammersley in [9], where amongother results, the fluctuations of the free case (no wall) were established in all dimensions (see also [7], where thisis discussed in more detail than in here).In [7], a solid flat wall is placed at the origin and its effect on the displacement of the surface with respect to itsinitial location at the origin is studied.In that reference, it is shown that in all dimensions the average height of the surface (say, at the origin) divergesto +∞ as time increases. This should be compared to the average absolute height of the surface at the origin inthe free case. In the latter case, that quantity is bounded in dimensions 3 and higher [9,7]. An effect of repulsionon the wall is thus established in those dimensions. Estimates on the speed of repulsion are obtained for a class ofnoise distributions (including the Gaussian case). These are comparable to estimates of the entropic repulsion forthe massless free field interacting with a flat wall (see [7] and references therein).Motivated by work on the entropic repulsion for the massless free field interacting with a wall with randomheights [1], we consider the same kind of wall here. In [1], estimates similar to those in [4–6] for the wall withfixed height case were obtained, showing in some cases an effect of the wall height distribution. We show theanalogous effect, with analogous quantitative estimates, for the same class of noise distributions considered in [7](see Theorem 3.1 below).Further studies on the massless free field interacting with a wall with random heights were carried on in [2,3].We refer again to [7] for other works on surfaces interacting with walls, in and out of equilibrium.In the next section we define precisely our model and describe the flat wall result of [7], which is related andrelevant to our main result. The latter is presented and argued in the following and final section. It was announcedpreviously in [8].2. The modelDenote by |i − j | the number of edges in a minimal path connecting i and j (we will use this definition not onlyfor Zd , but also for other graphs). LetP = {p(i, j)}i,j∈Zd be a symmetric stochastic matrix which satisfies p(i, j) =p(0, j − i) =: p(j − i) = p(i − j) (homogeneity) and p(j) = 0 for all |j | > v for some v (finiteness). Assumealso that P is truly d-dimensional: {j ∈ Zd : p(j) = 0} generates Zd . The weights p(i, j) can be interpreted astransition probabilities of a random walk on Zd ; denote by P its transition matrix and by pm(i, j) its m-steptransition probabilities. By homogeneity, pm(i, j) = pm(0, j − i) =: pm(j − i).Let E := {ε, εn(i), i ∈ Zd , n ∈ Z} be a family of i.i.d. integrable symmetric random variables with unboundedsupport. E represents the evolution noise variables.We next introduce the wall variables, giving the heights at each space coordinate. Consider the family of i.i.d.random variablesW= {W(i)}i∈Zd , independent of E . W(i) represents the height of the wall at site i.With a realization ofW fixed, the harness process interacting withW by exclusion is defined as follows.XWn (i) ={0 ∨ W(i), if n = 0,W(i) + (PXWn−1(i) + εn(i) − W(i))+, if n 1. (2.1)We allow W(i) to take the value −∞ (with positive probability), in which case the expression for n 1 in (2.1) isPXWn−1(i) + εn(i). (2.2)L.R.G. Fontes et al. / Ann. I. H. Poincaré – PR 42 (2006) 207–213 209Remark 2.1. Notice that XWn (i) so defined is nondecreasing in (the natural partial order for)W.The case whereW ≡ −∞, in which we denote XW by Y , is the free case introduced by Hammersley in [9].In that paper it is shown that EY 2n (0) is of order 1 in d  3. (Notice that under our assumptions on XW0 and ε,EYn(0) ≡ 0.)The case whereW≡ 0, in which we denote XW by Z, was studied in [7]. We now quote some of the results ofthat paper, which are directly related to our main result here.Theorem 2.1 (Part of Theorem 1.2 from [7]). Let Fε be the distribution function of ε and define the followingclasses of distribution functions:L−α :={F : F(x) c e−c′xα , x > 0, for some positive c, c′}, (2.3)L+α :={F : F(x) c e−c′xα , x > 0, for some positive c, c′}, (2.4)where F = 1 − F , andLα := L−α ∩L+α . (2.5)For d  3, there exist constants c and C that may depend on the dimension such that(i) if Fε ∈ Lα for some 1 α = 1 + d/2, thenc(logn)1α  EZn(0)C(logn)1α∨ 22+d ; (2.6)(ii) if Fε ∈ L1+d/2, thenc(logn)22+d  EZn(0) C(logn)22+d (log logn)d2+d . (2.7)3. ResultsIn the following, which is our main result, we obtain bounds on the average height of the wall at the origin as afunction of n, the number of iterations of the dynamics, in d  3. The average is taken with respect to the noise andwall variables. The bounds are similar to the corresponding ones in Theorem 2.1 above, and show an effect of thewall variables (to leading order, ignoring constants) when they have a heavy enough positive tail which is heavierthan the noise ones. This is the case when the noise variables are Gaussian and the wall ones are sub-Gaussian (i.e.,have distribution function belonging to Lθ with θ < 2).Theorem 3.1. Let d  3 and suppose that Fε(x) ∈ Lα and FW(x) ∈ Lθ . Then there exist c and C such thatc(logn)1α∨ 1θ  E XWn (0) C(An + (logn) 1θ), (3.1)whereAn ={(logn)1α∨ 22+d , if α = 1 + d2 ,(logn)22+d (log logn)dd+2 , if α = 1 + d2 .The lower bound in (3.1) is valid for α, θ > 0, and the upper bounds are valid for α > 1, θ > 0.210 L.R.G. Fontes et al. / Ann. I. H. Poincaré – PR 42 (2006) 207–213Proof.Lower bound. Let next Ŵ= {Ŵ (i)}i∈Zd , whereŴ (i) ={−∞, if W(i) < 0,0, if W(i) 0.It then follows that XWn XŴn . So, we need to obtain a lower bound for µn := EXŴn (0).Lemma 3.1. We have µn  c(logn)1/α , where c is a positive constant.Proof. Denote q = P(W(i)  0). With a slight abuse of notation we identify below Ŵ with the set {i ∈ Zd :Ŵ (i) = 0}. We have thenµn = E(XŴn (0) | 0 ∈ Ŵ)q + E(XŴn (0) | 0 /∈ Ŵ)(1 − q)= E[E((PXŴn−1(0) + εn(0))+ | Ŵ,0 ∈ Ŵ)]q + E[E(PXŴn−1(0) + εn(0) | Ŵ, 0 /∈ Ŵ)](1 − q)= EPXŴn−1(0) + E[E((PXŴn−1(0) + εn(0))− | Ŵ,0 ∈ Ŵ)]q=PEXŴn−1(0) + E[E((−PXŴn−1(0) + εn(0))+ | Ŵ,0 ∈ Ŵ)]q PEXŴn−1(0) + G(PEXŴ0n−1(0))q (3.2)= µn−1 + G(PEXŴ0n−1(0))q, (3.3)where a− := a+ − a, G(x) = E(ε − x)+, Ŵ0 = Ŵ∪ {0}, and (3.2) is due to Jensen’s inequality.We want now to estimate EXŴ0n (j) in terms of EXŴn (j). Consider the processes XŴ0n , XŴn and Yn = X∅n (freeprocess), all coupled together by using the same εk(j). We have that XŴ0n (j) XŴn (j) Yn(j) for all j . So, ifj = 0, using (2.1), (2.2) and the fact that for a  c it holds (a + b)+ − (c + b)+  a − c, we getXŴ0n (j) − XŴn (j)∑k∈Zdp(j, k)(XŴ0n−1(k) − XŴn−1(k)). (3.4)For j = 0 we haveXŴ0n (0) − XŴn (0)∑k∈Zdp(0, k)(XŴ0n−1(k) − XŴn−1(k))+ εn(0)− +PY−n−1(0) (3.5)(here we used the fact that for a  c g it holds (a + b)+ − (c + b) a − c + b− + g−). Iterating (3.4) and (3.5),one can get thatXŴ0n (j) − XŴn (j)n∑m=1pm−1(j)(εn−m+1(0)− +PY−n−m(0)). (3.6)Note that, as d  3, random walk with transition matrix P is transient and also EY−n (0) const, so (3.6) impliesthatEXŴ0n (j) − EXŴn (j) C0 (3.7)for some C0 > 0, for all n, i, and j .So, µn satisfiesµn  µn−1 + G˜(µn−1) (3.8)L.R.G. Fontes et al. / Ann. I. H. Poincaré – PR 42 (2006) 207–213 211where G˜(x) = G(x + C0)q . A lower bound of O((logn)1/α) for EXWn (0) then follows as in the proof of Theo-rem 3.1 and Corollary 3.3 in [7]. We derive next a lower bound of O((logn)1/θ ). Let µWn (i) = E(XWn (i) |W). As the dynamics of the processcan be re-written asXWn (i) ={0 ∨ W(i), if n = 0,PXWn−1(i) + εn(i) +(W(i) −PXWn−1(i) − εn(i))+, if n 1, (3.9)we haveµWn (i) =PµWn−1(i) + E((W(i) −∑j∈Zdp(i, j)XWn−1(j) − εn(i))+|W)PµWn−1(i) +(W(i) −PµWn−1(i))+. (3.10)ForW fixed, and i ∈ Zd letνWn (i) ={0 ∨ W(i), if n = 0,PνWn−1(i) +(W(i) −PνWn−1(i))+, if n 1. (3.11)We then have (since x + (a − x)+ is nondecreasing in x for all a) that µWn (i) νWn (i) for allW, n, i.Let next W˜= {W˜ (i)}i∈Zd , whereW˜ (j) ={W(j), if W(j) 0,−∞, if W(j) < 0.It follows that νWn (i) νW˜n (i) for allW, n, i.We will estimate νW˜n (i). Let us decompose W˜ in the following way. W˜= W˜i ∨ W˜i , with W˜i = {W˜i(j)}j∈Zdand W˜i = {W˜ i(j)}j∈Zd , whereW˜ i(j) ={W˜ (i), if i = j ,−∞, if i = j , W˜i(j) ={−∞, if i = j ,W˜ (i), if i = j . (3.12)Lemma 3.2. For all n and j it holdsνW˜in (j) ∨ νW˜in (j) νW˜n (j) νW˜in (j) + νW˜in (j). (3.13)Proof. We prove the lemma by induction. For n = 0 (3.13) is evident.Suppose (3.13) holds for n − 1. For j = i, we haveνW˜n (j) =PνW˜n−1(j) +(W˜ (j) −PνW˜n−1(j))+,νW˜in (j) =PνW˜in−1(j) +(W˜ (j) −PνW˜in−1(j))+,andνW˜in (j) =PνW˜in−1(j).Note that induction assumption implies PνW˜in−1(j) PνW˜n−1(j) and PνW˜in−1(j) PνW˜n−1(j). So, νW˜in (j) νW˜n (j),and, as x + (a − x)+ is increasing, νW˜in (j) νW˜n (j). Also by induction assumption,PνW˜n−1(j)PνW˜in−1(j) +PνW˜i (j ).n−1212 L.R.G. Fontes et al. / Ann. I. H. Poincaré – PR 42 (2006) 207–213As (a − x)+ is decreasing, we have(W˜ (j) −PνW˜n−1(j))+  (W˜ (j) −PνW˜in−1(j))+.Thus,νW˜n (j) νW˜in (j) + νW˜in (j).The case j = i is similar. Let us now estimate PνW˜0n (0). We suppose that W(0) =: W > 0, otherwise νW˜0n ≡ 0. We haveνW˜0n (i) =W, if i = 0, n = 0,0, if i = 0, n = 0,PνW˜0n−1(0) +(W −PνW˜0n−1(0))+, if i = 0, n 1,PνW˜0n−1(i), if i = 0, n 1.(3.14)It follows readily by induction that νW˜0n (i) W for all i, n. Thence, we have νW˜0n (0) ≡ W . It is now readilyverified by induction that for i = 0νW˜0n (i) = Wn∑k=1p{0}k (i,0), (3.15)where, for any j , p{0}k (j,0) is the probability that the random walk with transition matrix P starting from j returnsto 0 for the first time at step k. It follows thatPνW˜0n (0) = Wn+1∑k=1p{0}k (0,0) aW, (3.16)where a =∑∞k=1 p{0}k (0,0) < 1, since d  3.By (3.13) and (3.16), we haveνW˜n (0)PνW˜n−1(0) +((1 − a)W −PνW˜′n−1(0))+, (3.17)where W˜′ = {W˜ ′(i)}i∈Zd , with W˜ ′(i) = W˜ (i) if i = 0, and W˜ ′(0) independent of W˜. Taking now expectations withrespect to W˜, W˜ ′(0), and applying Jensen’s inequality, we getEνW˜n (0) EPνW˜n−1(0) + G(EPνW˜′n−1(0))=PEνW˜n−1(0) + G(PEνW˜′n−1(0))=PEνW˜n−1(0) + G(PEνW˜n−1(0))= EνW˜n−1(0) + G(EνW˜n−1(0)), (3.18)where G(x) = E((1 − a)W − x)+, and we have used the equidistribution of W˜ and W˜′, and the translation invari-ance of the joint distribution of W˜ and E .We then see that νn ≡ EνW˜n (0) is of the same form as (3.3) in [7], and a lower bound of O((logn)1/θ ) for νnfollows as in the proof of (3.4) in [7].Upper bound. As in [7], in order to obtain an upper bound, we compare the wall process with a free processstarted sufficiently high. Let XW,rnn and Y rnn have the same evolution as XWn (resp. Yn), but XW,rn0 (i) = rn ∨W(i),Yrn0 (i) ≡ rn. Let Rn = max{W(i): |i| vn} (recall that v is a range of P), andan ={2K((logn)1α∨ 22+d + (logn) 1θ ), if α = 1 + d2 ,2K((logn)22+d (log logn)dd+2 + (logn) 1θ ), if α = 1 + d . (3.19)2L.R.G. Fontes et al. / Ann. I. H. Poincaré – PR 42 (2006) 207–213 213Note that P(Rn > K(logn) 1θ ) nc1−c2Kθ . Take rn = an/2. We haveP(XWn (0) an) P(XW,rnn (0) an)= P(XW,rnn (0) an,XW,rnn (0) = Y rnn (0))+ P(XW,rnn (0) an,XW,rnn (0) = Y rnn (0)) P(Y rnn (0) an)+ P(XW,rnn (0) = Y rnn (0)). (3.20)In Section 5 of [7] it was shown that P(Y rnn (0) an) knc3−c4K . As for P(XW,rnn (0) = Y rnn (0)), note that XW,rnn (0)and Y rnn (0)) can be different if either if Rn > rn, or if it occurs{Yrnl (j) <Rn for some (l, j) with l  n, |j | v(n − l)}.We have thenP(Yrnl (j) <Rn) P(Yrnl (j) < K(logn)1θ)+ P(Rn > K(logn) 1θ ) P(Yl(j) < K(logn)1θ − rn)+ nc1−c2Kθ P(Yl(j) > rn − K(logn) 1θ)+ nc1−c2Kθ knc5−c6Kθ∧1 , (3.21)where θ ∧ 1 = min{θ,1}, andP(XW,rnn (0) = Y rnn (0)) nc1−c2Kθ +n∑l=0∑|j |v(n−l)knc5−c6Kθ∧1  k′nc7−c8Kθ∧1 .So,P(XWn (0) an) k∗nc′−c′′Kθ∧1and, by taking K large enough, this implies that EXWn (0) C2K an (see end of Section 6 of [7] for the reasoning ina similar situation). References[1] D. Bertacchi, G. Giacomin, Enhanced interface repulsion from quenched hard-wall randomness, Probab. Theory Relat. Fields 124 (2002)487–516.[2] D. Bertacchi, G. Giacomin, On the repulsion of an interface above a correlated substrate, in: Proceedings of 6th Brazilian School ofProbability, Bull. Braz. Math. Soc. (N.S.) 34 (3) (2003) 401–415.[3] D. Bertacchi, G. Giacomin, Wall repulsion and mutual interface repulsion: an harmonic crystal model in high dimensions, StochasticProcess. Appl. 110 (2004) 45–66.[4] E. Bolthausen, J.-D. Deuschel, G. Giacomin, Entropic repulsion and the maximum of the two-dimensional harmonic crystal, Ann.Probab. 29 (4) (2001) 1670–1692.[5] E. Bolthausen, J.-D. Deuschel, O. Zeitouni, Entropic repulsion of the lattice free field, Comm. Math. Phys. 170 (2) (1995) 417–443.[6] J.-D. Deuschel, Entropic repulsion of the lattice free field. II. The 0-boundary case, Comm. Math. Phys. 181 (3) (1996) 647–665.[7] P.A. Ferrari, L.R.G. Fontes, B. Niederhauser, M. Vachkovskaia, The serial harness interacting with a wall, Stochastic Process. Appl. 114(2004) 175–190.[8] L.R.G. Fontes, M. Vachkovskaia, A. Yambartsev, Entropic repulsion on a rarefied wall, in: Proceedings of the Conference “Discrete RandomWalks 2003”, Discrete Math. Theor. Comput. Sci. AC (2003) 105–112.[9] J.M. Hammersley, Harnesses, in: Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability, vol. III A, 1965/66, pp. 89–117.

Repulsion of an evolving surface on walls with random heights

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