FUNAKI, Tadahisa

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TitleA scaling limit for weakly pinned Gaussian random walks(Proceedings of RIMS Workshop on Stochastic Analysis andApplications)Author(s)FUNAKI, TadahisaCitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2008), B6: 97-109Issue Date2008-01URL http://hdl.handle.net/2433/174218RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Ko^kyu^roku Bessatsu B6(2008), 97{109A scaling limit for weakly pinnedGaussian random walksByTadahisa Funaki AbstractThe aim of this paper is threefold. First, we review the results of [1] on a scaling limit forweakly pinned Gaussian random walks. Secondly, two dierent proofs, one based on the FKGinequality and the other relying on the renewal theory, are given for an exponential decayestimate on certain probabilities related to the random walks. This plays a crucial role toestablish the large deviation principle. Finally, randomly pinned random walks are introduced.x 1. Model and resultsIn this section, we introduce the d-dimensional Gaussian random walks with pinningat the origin 0. Then the results of [1] on a macroscopic scaling limit for such randomwalks are reviewed. See Section 6 of [3] for further related results. The paper [4]discusses the same problem in a critical situation for the pinned Wiener measures withweak self potentials.x 1.1. Weakly pinned Gaussian random walksFor   0, let  = (i)i2DN be the Markov chain on Rd with transition probabilityP (x; dy) =1Z(x)e jx yj2=2  0(dy) + dy; x; y 2 Rd;where DN = f0; 1; 2; : : : ; Ng; N 2 N and Z(x) =  e jxj2=2+(2)d=2 is the normalizingconstant. If  = 0, the Markov chain  is the d-dimensional Brownian motion viewedat integer times. The parameter  represents the strength of the pinning at the origin2000 Mathematics Subject Classication(s): 60K35, 60F10, 82B41Supported in part by the JSPS Grants 17654020 and (A) 18204007.Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914,Japan. e-mail: funaki@ms.u-tokyo.ac.jpc 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.98 Tadahisa Funaki0. In [1], the random walks weakly pinned at a general subspace M of Rd are studied,but in the present paper we take M = f0g for simplicity.Given a 2 Rd, the distribution on (Rd)DN of the Markov chain  starting at aN isdenoted by a;F;N , i.e.,a;F;N (d) =1Za;F;Ne HN ()aN (d0)NYi=1  0(di) + di;(1.1)where Za;F;N is the normalizing constant andHN () =12N 1Xi=0ji+1   ij2:Given b 2 Rd, the distribution of the Markov chain conditioned to arrive at bN isdenoted by a;b;N ; in other words, a;b;N is the probability measure given by the formula(1.1) with the last factor ( 0(dN ) + dN ) replaced by bN (dN ) and Za;F;N by a newnormalizing constant Za;b;N , respectively.We also consider the Markov chains moving only on the upper half space Rd+ =Rd 1  [0;1) of Rd. Namely, for a; b 2 Rd+, we consider the conditional distributionsa;F;;+N and a;b;;+N of a;F;N and a;b;N , respectively, on the event f;i 2 Rd+ forall i 2 DNg = (Rd+)DN. The corresponding normalizing constants are denoted byZa;F;;+N and Za;b;;+N , respectively. These Markov chains are dened under the presenceof a wall at @Rd+.The following table shows the dierence of these four measures in short:No Wall Wall at @Rd+Pinned at N a;b;N a;b;;+NFree at N a;F;N a;F;;+NTable 1. Distributions of Markov chainsx 1.2. Results1.2.1. Free energies and phase transitionsWe dene the free energies associated with the weakly pinned random walks by theWeakly pinned random walks 99thermodynamic limits under the absence or presence of a wall: = limN!11NlogZ0;0;NZ0;0N = limN!11NlogZ0;F;NZ0;FN!;;+ = limN!11NlogZ0;0;;+NZ0;0;+N = limN!11NlogZ0;F;;+NZ0;F;+N!:We have chosen a = b = 0 and the partition functions (normalizing constants) in thedenominators are dened without pinning; for example, Z0;0N = Z0;0;0N (i.e.,  = 0). Onecan show that the limits exist and are independent of the conditions at i = N (pinnedor free), and  > ;+  0 for all   0 unless  = 0. The random walks exhibit thephase transition (localization/delocalization transition or recurrence/transience transi-tion) depending on the strength  of the pinning as is summarized in the next table:d  3 :9 c > 0 s.t.Pinning transition  > c )  > 0 (positive recurrent regime)(Absence of wall) 0    c )  = 0 (null recur./transient regime)d = 1; 2 : c = 0Wetting transition d  1 :9 +c > c s.t.(Presence of wall)  > +c ) ;+ > 0 (positive recurrent regime)0    +c ) ;+ = 0 (null recur./transient regime)Table 2. Phase transitionsExplicit formulae determining  and c are found in Section 2.2.1, while those for;+ and +c are in Section 2.2.2. Asymptotic behaviors of  or ;+ as  # c or  # +ccan be studied and, especially, the critical exponents associated with the free energiescan be computed explicitly, see [1].1.2.2. Sample path large deviation principleWe are interested in a macroscopic behavior of the Markov chains. Let hN =fhN (t); t 2 Dg be the macroscopic path of the Markov chain determined from themicroscopic one  under the scaling dened through a polygonal approximation of hN (i=N) = i=Ni2DN :hN (t) =[Nt] Nt+ 1N[Nt] +Nt  [Nt]N[Nt]+1; t 2 D;where D = [0; 1]. Then, as N !1, the sample path large deviation principle holds forhN under each of the four distributions a;b;N , a;b;;+N , a;F;N and a;F;;+N introduced in100 Tadahisa FunakiSection 1.1 on the space C = C(D;Rd) equipped with the uniform topology. The speedof the large deviation principle is always N and the unnormalized rate functionals aregiven by a;b;, a;b;;+, a;F; and a;F;;+, respectively, of the form(h) =12ZDj _h(t)j2dt  ft 2 D;h(t) = 0g;(1.2)for h 2 H1(D;Rd), where  =  for  = a;b;, a;F; and  = ;+ for  = a;b;;+,a;F;;+, j  j stands for the Lebesgue measure on D and H1(D;Rd) is the usual Sobolevspace.The large deviation principle immediately implies the concentration for each of thedistributions N = a;b;N ; a;b;;+N ; a;F;N and a;F;;+N :limN!1N dist1(hN ;H)  = 1;(1.3)for every  > 0, where H = fh;minimizers of g and dist1 denotes the distance on Cunder the uniform norm k  k1.There are at most two possible minimizers of :Functionals a;b; a;b;;+ a;F; a;F;;+Possible minimizers h; h^ h; h^+ hF ; h^F hF ; h^F;+Table 3. Possible minimizersHere, the functions h; h^; hF and h^F are dened as follows:h(t) = (1  t)a+ tb; h^(t) =8><>:(t1   t)a=t1; t 2 [0; t1];0; t 2 [t1; 1  t2];(t+ t2   1)b=t2; t 2 [1  t2; 1];hF (t) = a; h^F (t) =((t1   t)a=t1; t 2 [0; t1];0; t 2 [t1; 1];for t 2 D, where t1 = jaj=p2 and t2 = jbj=p2. The functions h^+ and h^F;+ aredened similarly to h^ and h^F , respectively, with  replaced by ;+. The functions hand hF are linear, i.e., h linearly interpolates between a and b, while hF stays at a.On the other hand, the functions h^; h^F ; h^+ and h^F;+ visits 0 to gain a reward from thesecond term of (1.2), paying a penalty for the rst term.In particular, if the minimizer of  is unique, we see from (1.3) that the law oflarge numbers holds for hN under N and the limit is the unique minimizer.Weakly pinned random walks 1011.2.3. Scaling limits in a critical situationOf particular interest is the critical situation that the two possible minimizers areactually the minimizers of  simultaneously: a;b;(h) = a;b;(h^) and others. Theresults, summarized in the next table, are supercially independent of the presence orabsence of a wall, but depend on the conditions at i = N and the dimension d of thespace.LimitsPinned at N d = 1 h^ (or h^+)(a;b;N ; a;b;;+N ) d = 2 Coexistenced  3 hFree at N d = 1 Coexistence(a;F;N ; a;F;;+N ) d  2 hFTable 4. Scaling limits at criticalityHere, \Coexistence" means that the distribution on C of hN under N converges weaklyto the superposition of positive masses at two minimizers as N !1. See [1] for details.The central limit theorem is also established in [1] for the rst or last hitting timesof the Markov chains  at 0.x 2. Large deviation type estimatesThe following estimates play a crucial role to establish the sample path large de-viation principle for hN under a;b;N ; a;b;;+N ; a;F;N and a;F;;+N , cf. Section 4 of [1]. Infact, the proof of the large deviation principle can be reduced to the following roughlarge deviation estimates under the 0-boundary conditions (i.e., a = b = 0).Proposition 2.1. For every  > 0, there exists C; c > 0 independent of N suchthatN (khNk1  )  Ce cNfor N = 0;0;N ; 0;0;;+N ; 0;F;N and 0;F;;+N .We give two dierent proofs of this proposition. Section 2.1, based on a stochasticdomination for Euclidean norms, is for two measures 0;0;N and 0;F;N , i.e., under theabsence of a wall; also based on a stochastic domination, [1] covered all four measuresby viewing conditional distributions. The method of Section 2.2, based on a renewaltheory and announced in Remark 4.1-(2) of [1], is applicable to all four measures.102 Tadahisa Funakix 2.1. Proof based on a stochastic dominationLet R : (Rd)DN ! (R+)DN be a mapping dened by R = x with x = (xi =jij)i2DN for  = (i)i2DN . The space (R+)DN is equipped with a natural partial orderx  y for x = (xi)i2DN and y = (yi)i2DN dened by xi  yi for every i 2 DN . For twoprobability measures 1 and 2 on (R+)DN , we say that 2 stochastically dominates 1(written as 1  2) if E1 [F ]  E2 [F ] holds for all bounded non-decreasing functionsF on (R+)DN .Lemma 2.1. We have the stochastic dominations:0;0;N R 1 < 0;0N R 1;0;F;N R 1 < 0;FN R 1:Proof. Since x = (xi)i2DN is a d-dimensional Bessel process viewed at integertimes under 0;FN R 1, it enjoys the FKG inequality; see Proposition 5.6 of [5]. However,the probability measure 0;F;N R 1 is given by the weak limit of a sequence of certainprobability measures fkgk2N having non-increasing densities with respect to 0;FN R 1.Therefore, the FKG inequality for 0;FN  R 1 implies the stochastic domination k <0;FN R 1 for every k, and this concludes 0;F;N R 1 < 0;FN R 1 by taking the limitk !1.On the other hand, the density of 0;0N R 1 satises the so-called Holley's condition(since we may only consider the density of the joint distributions of the d-dimensionalBessel processes at integer times under the condition xN = 0) and thus the FKGinequality holds for 0;0N  R 1. The rest of the proof for 0;0;N  R 1 < 0;0N  R 1 issimilar.Proof of Proposition 2.1 for 0;0;N and 0;F;N . The proof for 0;0;N is immediate:0;0;N (khNk1  )  0;0N (khNk1  )  Ce cN ;where we have applied Lemma 2.1 for the rst inequality and then the standard largedeviation principle for 0;0N , under which  is a Brownian bridge viewed at integer times.The proof for 0;F;N is similar.x 2.2. Proof based on a renewal theoryWe give the proof only for 0;0;N and 0;0;;+N , the case where the Markov chains arepinned at N , since the free case can be discussed in a parallel way.2.2.1. The pinned case without a wallWeakly pinned random walks 103For  > 0 and   0, setpn() =Z0;0;nZ0;0n0;0;n (khnk1  ):Then, by expanding the measure as in the proof of Lemma 2.1 (or (3.3)) of [1], we havepn() =0;0n (khnk1  )+n 1Xi=1Z0;0i Z0;0;n iZ0;0n0;0i (khnk1;[0;i=n]  )0;0;n i (khnk1;[i=n;1]  )=p0n() +n 1Xi=1Z0;0i;n ip0i (n=i)pn i(n=(n  i));whereZ0;0i;n i Z0;0i Z0;0n iZ0;0n=(2n)d=2(2i)d=2(2(n  i))d=2 :The last line for pn() and the last identity for Z0;0i;n i follow from fkhnk1;[0;i=n] g = fkhik1  n=ig, fkhnk1;[i=n;1]  g = fkhn ik1  n=(n   i)g and Z0;0n =(2)dn=2=(2n)d=2, respectively. Since p0i (n=i)  p0i () and pn i(n=(n i))  pn i(),the above calculation leads to the inequality:un  bn +nXi=0aiun i; n  0;for three sequences fun  un(x; ); an  an(x; ); bn  bn(x)g1n=0 dened byun =xn(2n)d=2pn();bn =xn(2n)d=2p0n() = un(x; 0);an =xn(2n)d=2p0n() = bn;for n  1 and u0 = a0 = b0 = 0, respectively, where x  0.Compared with the solution fvn  vn(x; )g1n=0 of the renewal equation:vn = bn +nXi=0aivn i; n  0;(2.1)it is immediate to see that 0  un  vn by induction, since the coecients ai arenon-negative.104 Tadahisa FunakiTo nd the asymptotic behavior of vn(x; ) as n!1, dene an increasing functionf(x) for x  0 byf(x) =1Xn=1xn(2n)d=20;0n (khnk1  ) 1Xn=1bn(x)!:Then we have the following lemma.Lemma 2.2. (1) Let x be the radius of convergence of f . Then x > 1.(2) For  > 1=f(x ) 2 [0;1), we havelimn!1 vn(x; ) =12xf 0(x)(> 0);where x 2 (0; x) is the unique solution of f(x) = 1=.(3) If f(x ) < 1 and 0 <   1=f(x ), the sequence fvn(x   ; ) > 0gn2N isbounded for every  2 (0; x).Proof. The assertion (1) follows from the fact that 0;0n (khnk1  ) decays expo-nentially fast in n. For (2), sinceP1n=0 an(x; ) = 1 under the choice of x = x, therenewal theory (Chapter XIII of [2]) shows thatvn P1n=0 bnP1n=0 nan=12xf 0(x);as n ! 1, where  means that the ratio of both sides tends to 1. The assertion (3)follows again by applying the renewal theory to vn(x  ; ) noting thatP1n=0 an(x ; ) = f(x   ) < 1.Let g be the function dened byg(x) =1Xn=1xn(2n)d=2;for x  0. It is shown in [1] that the free energy  is given by  =   log x, wherex 2 (0; 1] is the unique solution of g(x) = 1= for  > c := 1=g(1 ) and x = 1 for0 <   c. Since 0 < f(x) < g(x) for x > 0 (as long as f(x) <1), choosing x 2 (0; x)as in Lemma 2.2 for  > 1=f(x ) and x 2 (1; x) arbitrarily for   1=f(x ) (iff(x ) <1), we have0 < x < x(2.2)for all  > 0. Under the choice of x = x, we have0;0;n (khnk1  ) = (2n)d=2(x) nun(x; )Z0;0;nZ0;0n 1;Weakly pinned random walks 105and thereforelim supn!11nlog0;0;n (khnk1  )   log x + lim supn!11nlog vn(x; )   log x   log x < 0;from Lemma 2.2 and (2.2). This completes the proof of Proposition 2.1 for 0;0;N .2.2.2. The pinned case with a wallFor  > 0 and   0, setp;+n () =Z0;0;;+nZ0;0;+n0;0;;+n (khnk1  ):Then, an identity similar to that for pn() holds for p;+n () by replacing Z0;0i;n i withZ0;0;+i;n i Z0;0;+i Z0;0;+n iZ0;0;+n:Since Z0;0;+n = Z0;0n =n, this provesu+n  b+n +nXi=0a+i u+n i; n  0;for three sequences fu+n  u+n (x; ); a+n  a+n (x; ); b+n  b+n (x)g1n=0 dened byu+n =xnn(2n)d=2p;+n ();b+n = u+n (x; 0);a+n = b+n ;for n  1 and u+0 = a+0 = b+0 = 0, respectively, where x  0. We dene two increasingfunctions f+(x) and g+(x) for x  0 byf+(x) =1Xn=1xnn(2n)d=20;0;+n (khnk1  ) 1Xn=1b+n (x)!;g+(x) =1Xn=1xnn(2n)d=2:It is shown in [1] that the free energy ;+ is given by ;+ =   log x;+, where x;+ 2(0; 1] is the unique solution of g+(x) = 1= for  > +c := 1=g+(1) and x;+ = 1 for106 Tadahisa Funaki0 <   +c . Since 0 < f+(x) < g+(x) for x > 0 (as long as f+(x) <1), the rest of theproof of Proposition 2.1 for 0;0;;+N is similar to Section 2.2.1.2.2.3. Further application of the renewal theoryAs another application of the renewal theory, we give the proof of the followingproposition. This proposition is shown in [1] as a direct consequence of the preciseasymptotics for the ratios of partition functions, see Remark 2.1-(2) of [1]. In a sense,we repeat the argument there from a slightly dierent view point.Proposition 2.2. If  > +c , we have that0;0;N (i 2 Rd+ for all i 2 DN )  Ce N( ;+);as N !1, where C = xg0(x)=g(x;+).Proof. As in Section 2.2.1, we haveqn = q0n +n 1Xi=1Z0;0i;n iq0i qn i;forqn =Z0;0;nZ0;0n0;0;n (i 2 Rd+ for all i 2 Dn):Therefore, three sequences fvn  vn(x; ); an  an(x; ); bn  bn(x)g1n=0 dened byvn =xn(2n)d=2qn;bn = vn(x; 0);an = bn;with v0 = a0 = b0 = 0 fulll the renewal equation (2.1), where x  0. Sinceq0n = 0;0n (i 2 Rd+ for all i 2 Dn) =1n;we see that1Xn=1an = 1Xn=1xnn(2n)d=2= g+(x) = 1under the choice of x = x;+ = e ;+. Thus, the renewal theory implies thatvn  12x;+(g+)0(x;+)=12g(x;+);Weakly pinned random walks 107as n!1. The conclusion follows fromZ0;0;nZ0;0n (2n)d=22xg0(x)en;which is shown in Proposition 2.2 of [1].x 3. Pinning model in a random environmentFinally in this section, we propose a model of Gaussian random walks perturbedby an attractive force toward a subspace of Rd, which is chosen randomly at every timei. The model is stated for the free case, but the pinned case can be discussed similarly.Related random polymer models are studied in Giacomin [6], for instance, see case Ain Section 1.1 of [6].Let  = ()d=1 2 (0;1)d be given;  represents the direction-dependent strengthof the pinning. Set(a) =Y2a and a(dx) =Y2a0(dx)Y=2adx ;for a 2  and x = (x)d=1 2 Rd, where  = fa; non-empty subsets of f1; : : : ; dgg. Themeasure a on Rd is concentrated on its subspace:Ma = fx 2 Rd;x = 0 for  2 ag:Let p be an arbitrary probability measure on  such that p(a) > 0 for each a 2 . Wedenote  = DN .Denition 3.1. We dene a (quenched) pinning model in an environment ! =(a1; : : : ; aN ) 2  bya;F;!N (d) =1Za;F;!Ne HN ()aN (d0)NYi=1(ai)p(ai)ai(di) + di;where a 2 Rd and Za;F;!N is the normalizing constant.Under a;F;!N , the pinning occurs randomly in i to the subspace Mai with thestrength (ai)=p(ai). Denoting a = (a)d=1 2 Rd, let us consider the distributiona;F;N on RDN of the one-dimensional Markov chains dened by (1.1) for each 1   d. We then dene the distribution on (Rd)DN by the producta;F;N (d) =dY=1a;F;N (d):108 Tadahisa FunakiProposition 3.1. We have thata;F;N (d) = E[Z!a;F;!N (d)];where E stands for the expectation on the probability space (; pN ); Z! = Za;F;!N =Za;F;Nand Za;F;N =Qd=1 Za;F;N .Proof. Rewrite a;F;N asa;F;N (d) =1Za;F;Ne HN ()aN (d0)NYi=1dY=1 0(di ) + di;and expand the measure asdY=1 0(dx) + dx=Xa0f1;::: ;dg(a0)a0(dx)=Xa6=;(a)a(dx) + dx=Xa2 (a)a(dx) + p(a)dx;for x = (x)d=1 2 Rd, where (;) = 1 and ;(dx) = dx. Thus, we havea;F;N (d) =1Za;F;Ne HN ()aN (d0)NYi=1Xai2 (ai)ai(di) + p(ai)di=1Za;F;NX!=(a1;::: ;aN )2 NYi=1p(ai)!e HN () aN (d0)NYi=1(ai)p(ai)ai(di) + di=E"Za;F;!NZa;F;Na;F;!N (d)#:This concludes the proof.An asymptotic behavior of hN under the annealed measure a;F;N is not dicultyto study. In fact, since it is a product measure, the large deviation principle holds forhN 2 C under a;F;N and the corresponding unnormalized rate functional is given by(h) =12ZDj _h(t)j2 dt dX=1ft 2 D;h(t) = 0g:Weakly pinned random walks 109References[1] E. Bolthausen, T. Funaki and T. Otobe, Concentration under scaling limits for weaklypinned Gaussian random walks, preprint.[2] W. Feller, An Introduction to Probability Theory and its Applications, vol. 1, Seconded., John Wiley and Sons, 1966.[3] T. Funaki, Stochastic Interface Models, In: Lectures on Probability Theory and Statistics,Ecole d'Ete de Probabilites de Saint-Flour XXXIII - 2003 (ed. J. Picard), pp. 103{274,Lect. Notes Math., 1869 (2005), Springer.[4] T. Funaki, Dichotomy in a scaling limit under Wiener measure with density, preprint.[5] T. Funaki, Y. Hariya and M. Yor, Wiener integrals for centered powers of Besselprocesses, I, to appear in Markov Proc. Relat. Fields.[6] G. Giacomin, Random Polymer Models, book, 2006.

A scaling limit for weakly pinned Gaussian random walks (Proceedings of RIMS Workshop on Stochastic Analysis and Applications)

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A scaling limit for weakly pinned Gaussian random walks (Proceedings of RIMS Workshop on Stochastic Analysis and Applications)

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