ON THE DELOCALIZED PHASE OF THE RANDOM PINNING MODEL

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1 O THE DELOCALIZED PHASE OF THE RADOM PIIG MODEL JEA-CHRISTOPHE MOURRAT Abstract. We cosider the model of a directed polymer pied to a lie of i.i.d. radom charges, ad focus o the iterior of the delocalized phase. We first show that i this regio, the partitio fuctio remais bouded. We the prove that for almost every eviromet of charges, the probability that the umber of cotact poits i [0,] exceeds clog teds to 0 as teds to ifiity. The proofs rely o recet results of [BGdH0, CdH0].. Itroductio Let τ = (τ i ) i be a sequece such that τ 0 = 0 ad (τ i+ τ i ) i 0 areidepedet ad idetically distributed radom variables with values i = {,2,...}. Let P be the distributio of τ, E the associated expectatio, ad K() = P[τ = ]. We assume that there exists α 0 such that (.) log K() log (+α). As a example, oe ca thik about the sequece τ as the sequece of arrival times at 0 of a oe-dimesioal simple radom walk (ad i this case, α = /2). I a slight abuse of otatio, we will look also at the sequece τ as a set, ad write for istace τ istead of i : = τ i. Let ω = (ω k ) k be idepedet ad idetically distributed radom variables. We write P for the law of ω, ad E for the associated expectatio. We will refer to ω as the eviromet. We assume that the ω k are cetred radom variables, ad that they have expoetial momets of all order. Let β 0,h 0, ad. (expectatio E β,h,ω We cosider the probability measure P β,h,ω as the followig Gibbs trasformatio of the measure P : dp β,h,ω dp (τ) = exp ( (βω k h) {k τ} ) { τ}. ) which is defied I the above defiitio, β ca be thought of as the iverse temperature, h as the disorder bias, ad Z β,h,ω is a ormalizatio costat called the partitio fuctio, [ ( ) ] = E exp (βω k h) {k τ} { τ}. At the expoetial scale, the asymptotic behaviour of the partitio fuctio is captured by the free eergy f(β,h) defied as f(β,h) = lim +. logzβ,h,ω Superadditivity of the partitio fuctio implies that this limit is well defied almost surely, ad that it is determiistic (see for istace [G, Theorem 4.]). Assumptio (.) implies that f(β,h) 0. It is ituitively clear that the free eergy ca become strictly positive oly if the set τ [0,] is likely to cotai may poits uder the measure P β,h,ω. We thus say that we are i the localized phase

2 2 JEA-CHRISTOPHE MOURRAT if f(β,h) > 0, ad i the delocalized phase otherwise. Oe ca show [dh, Theorem.3] that for every β 0, there exists h c (β) 0 such that h < h c (β) localized phase, i.e. f(β,h) > 0, h h c (β) delocalized phase, i.e. f(β,h) = 0, ad moreover, the fuctio β h c (β) is strictly icreasig. 2. Statemet of the mai results We focus here o the iterior of the delocalized phase, that is to say whe h > h c (β). ote that, due to the strict mootoicity of the fuctio h c ( ), oe sits ideed i the iterior of the delocalized phase if oe fixes h = h c (β 0 ) ad cosiders ay iverse temperature β < β 0. By defiitio, the partitio fuctio is kow to grow subexpoetially i this regio. I [BS0, Remark p. 47], the authors ask whether the partitio fuctio remais bouded there. We aswer positively to this questio, ad ca i fact be slightly more precise. Theorem 2.. Let β 0 ad h > h c (β). For almost every eviromet, oe has + < +. = Remark. This result implies that, i the iterior of the delocalized phase, the ucostraied (or free) partitio fuctio,f is also almost surely bouded (i fact, teds to 0) as teds to ifiity. Ideed,,f is defied by [ ( )] = E exp (βω k h) {k τ},,f which is equal to [ ( + ) ] E exp (βω k h) {k τ} ;τ [, ] = { } = + = a.s. 0. Our secod result cocers the size of the set τ [0,], that we may call the set of cotact poits, uder the measure P β,h,ω. Let us write E, for the evet that τ [0,] > (where we write A for the cardial of a set A). Theorem 2.2. Let β 0 ad h > h c (β). For every ε > 0 ad for almost every eviromet, there exists ε,c ε > 0 such that for ay ε ad ay : P β,h,ω (E, ) C ε K() e (h hc(β) ε). I particular, for every costat c such that c > +α h h c (β) ad for almost every eviromet, oe has P β,h,ω (E,clog ) 0. To my kowledge, results of this kid were kow oly uder the averaged measure PP β,h,ω, ad with some restrictios o the distributio of ω due to the use of cocetratio argumets (see [GT05] or [G, Sectio 8.2]). I particular, i the iterior of the delocalized phase ad for almost every eviromet, the polymer itersects the piig lie less that the simple radom walk does.

3 O THE DELOCALIZED PHASE OF THE RADOM PIIG MODEL 3 It is worth comparig this result with the case whe radomess of the medium is abset, that is, whe β = 0. I this cotext, the distributio of the umber of cotact poits of the polymer forms a tight sequece as varies (see for istace [dh, Theorem 7.3]). It is oly atural to expect that a similar result holds true i the disordered case as well. Iterestigly, boudedess of the umber of cotact poits i the delocalized phase was recetly obtaied for a specific model of piig o a radom iterface with log-ragecorrelatiosi [BL], eve at criticality (h = h c (β)). I this work, the specific structure of the eviromet eables the authors to idetify the critical poit explicitly, a feature which makes the subsequet aalysis more tractable. 3. Proofs I this sectio, we preset the proofs of Theorems 2. ad 2.2. Although oe might thik at first that such a approach caot be of much help as far as the delocalized phase is cocered, we will rely o recet results obtaied i [BGdH0, CdH0], where the authors develop a large deviatios poit of view of the problem. Let us defie (3.2) = 0=l 0<l < <l i=0 K(l i+ l i )e (βω l i h). Our results are based o the followig fact, due to [CdH0], that holds both i the delocalized ad i the localized phases. Lemma 3.. For almost every eviromet, oe has lim sup + logfβ,h,ω = h c (β) h. Proof of Lemma 3.. Althoughthis result is otstated asapropositioi [CdH0], the authors give all the ecessary elemets to prove it. Ideed, we ca start from [CdH0, (3.)], which reads lim sup + logfβ,h,ω = h+s que (β;), where S que (β;z) is defied i [CdH0, (3.0)]. We the lear from [CdH0, (3.3)] that h c (β) = S que (β; ), so what remais to see is that S que (β; ) = S que (β;). The proof of this fact is first obtaied assumig further that the support of the distributio of ω 0 is fiite, see [CdH0, Lemma 3.2]. The geeral case, where oe assumes oly fiiteess of all expoetial momets, is cosidered i [CdH0, Sectio 3.3]. Oe ca start by observig that, sice z S que (β,z) is icreasig (i the wide sese), oe has S que (β; ) S que (β;). O the other had, it is show i step of the proof of [CdH0, Lemma 3.3] that S que (β;) A(β), where A(β) is defied i [CdH0, (3.2)]. Moreover, steps 2 to 4 of the proof of [CdH0, Lemma 3.3] are devoted to the justificatio of the fact that A(β) S que (β; ). It thus follows that S que (β;) S que (β; ), which fiishes the proof. Proof of Theorem 2.. The proof is close to [CdH0, Sectio 3.2]. We ca decompose Z the followig way : = + =0=l 0<l < <l = i=0 K(l i+ l i )e (βω l i h).

4 4 JEA-CHRISTOPHE MOURRAT A iterversio of sums the leads to + = + = =, ad Lemma 3. esures the almost sure covergece of the secod series whe h > h c (β). For a evet A, let us write Z β,h,ω (A) for the quatity [ ( ] E exp (βω k h) {k τ} ) { τ} ; A. I words, Z β,h,ω (A) is a partitio fuctio i which oe itegrates with respect to P oly o the evet A. I order to prove Theorem 2.2, we first give a refied versio of Theorem 2., which goes as follows. Propositio 3.2. Let β 0 ad h > h c (β). For every ε > 0 ad for almost every eviromet, there exist ε,c ε such that for ay ε : + = (E, ) C ε e (h hc(β) ε). Proof. We ca assume that ε < h h c (β). ote that, for ay ad 0, (E,0 ) = + = 0 0=l 0<l < <l = i=0 By a iterversio of sums, we obtai that + = (E,0 ) = +. = 0 By Lemma 3., there exists ε such that for every ε, e (h hc(β) ε/2), ad as a cosequece, for every 0 ε, oe has K(l i+ l i )e (βω l i h). + = Z β,h,ω (E,0 ) which implies the aouced claim. + = 0 e (h hc(β) ε/2), Proof of Theorem 2.2. ote that P β,h,ω (E, ) = Zβ,h,ω (E, ). The umerator ca be bouded from above usig Propositio 3.2. For the deomiator, oe ca use the boud K()e βω0 h, which proves the desired result.

5 O THE DELOCALIZED PHASE OF THE RADOM PIIG MODEL 5 Refereces [BL] Q. Berger, H. Lacoi. Sharp critical behavior for piig model i radom correlated eviromet. Preprit, arxiv: v (20). [BGdH0] M. Birker, A. Greve, F. de Hollader. Queched large deviatio priciple for words i a letter sequece. Probab. Theory Related Fields 48 (3-4), (200). [BS0] M. Birker, R. Su. Aealed vs queched critical poits for a radom walk piig model. A. Ist. Heri Poicaré Probab. Stat. 46 (2), (200). [CdH0] D. Cheliotis, F. de Hollader. Variatioal characterizatio of the critical curve for piig of radom polymers. Preprit, arxiv: v (200). [G] G. Giacomi. Radom polymer models. Imperial College Press (2007). [GT05] G. Giacomi, F.L. Toielli. Estimates o path delocalizatio for copolymers at selective iterfaces. Probab. Theory Related Fields 33 (4), (2005). [dh] F. de Hollader. Radom polymers. Ecole d été de probabilités de Sait Flour XXXVII, Lecture otes i Mathematics 974, Spriger (2009). Ecole polytechique fédérale de Lausae, istitut de mathématiques, statio 8, 05 Lausae, Switzerlad