Probab. Theory Relat. Fields (2016) 164:17–59 DOI 10.1007/s00440-014-0606-4

The continuum disordered pinning model Francesco Caravenna · Rongfeng Sun · Nikos Zygouras

Received: 19 June 2014 / Revised: 30 November 2014 / Published online: 17 December 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract Any renewal processes on N0 with a polynomial tail, with exponent α ∈ (0, 1), has a non-trivial scaling limit, known as the α-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for α ∈ 21 , 1 these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of R in a white noise random environment, with subtle features: • Any fixed a.s. property of the α-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment. • Nonetheless, the law of the CDPM is singular with respect to the law of the α-stable regenerative set, for almost every realization of the environment. The existence of a disordered continuum model, such  CDPM, is a manifestation  as the of disorder relevance for pinning models with α ∈ 21 , 1 .

F. Caravenna (B) Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 55, 20125 Milano, Italy e-mail: [email protected] R. Sun Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore e-mail: [email protected] N. Zygouras Department of Statistics, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected]

123

18

F. Caravenna et al.

Keywords Scaling limit · Disorder relevance · Weak disorder · Pinning model · Fell–Matheron topology · Hausdorff metric · Random polymer · Wiener Chaos expansion Mathematics Subject Classification

Primary 82B44; Secondary 82D60 · 60K35

1 Introduction We consider disordered pinning models, which are defined via a Gibbs change of measure of a renewal process, depending on an external i.i.d. random environment. First introduced in the physics and biology literature, these models have attracted much attention due to their rich structure, which is amenable to a rigorous investigation; see, e.g., the monographs of Giacomin [19,20] and den Hollander [13]. In this paper we define a continuum disordered pinning model (CDPM), inspired by recent work of Alberts et al. [4] on the directed polymer in random environment. The interest for such a continuum model is manifold: • It is a universal object, arising as the scaling limit of discrete disordered pinning models in a suitable continuum and weak disorder limit, Theorem 1.3. • It provides a tool to capture the emerging effect of disorder in pinning models, when disorder is relevant, Sect. 1.4 for a more detailed discussion. • It can be interpreted as an α-stable regenerative set in a white noise random environment, displaying subtle properties, Theorems 1.4, 1.5 and 1.6. Throughout the paper, we use the conventions N := {1, 2, . . .}, N0 := {0} ∪ N, and write an ∼ bn to mean limn→∞ an /bn = 1. 1.1 Renewal processes and regenerative sets Let τ := (τn )n≥0 be a renewal process on N0 , that is τ0 = 0 and the increments (τn −τn−1 )n∈N are i.i.d. N-valued random variables (so that 0 = τ0 < τ1 < τ2 < · · · ). Probability and expectation for τ will be denoted respectively by P and E. We assume that τ is non-terminating, i.e., P(τ1 < ∞) = 1, and K (n) := P(τ1 = n) =

L(n) , n 1+α

as n → ∞,

(1.1)

where α ∈ (0, 1) and L(·) is a slowly varying function [8]. We assume for simplicity that K (n) > 0 for every n ∈ N (periodicity complicates notation, but can be easily incorporated). Let us denote by C the space of all closed subsets of R. There is a natural topology on C, called the Fell–Matheron topology [15,24,25], which turns C into a compact Polish space, i.e. a compact separable topological space which admits a complete metric. This can be taken as a version of the Hausdorff distance (see Appendix A for more details).

123

The continuum disordered pinning model

19

Identifying the renewal process τ = {τn }n≥0 with its range, we may view τ as a random subset of N0 , i.e. as a C-value random variable (hence we write {n ∈ τ } :=  {τ = n}). This viewpoint is very fruitful, because as N → ∞ the rescaled set k k≥0 τ = N



τn N

 (1.2) n≥0

converges in distribution on C to a universal random closed set τ of [0, ∞), called the α-stable regenerative set ([16], [19, Thm. A.8]). This coincides with the closure of the range of the α-stable subordinator or, equivalently, with the zero level set of a Bessel process of dimension δ = 2(1 − α) (see Appendix A), and we denote its law by Pα . Remark 1.1 Random sets have been studied extensively [24,25]. Here we focus on the special case of random closed subsets of R. The theory developed in [16] for regenerative sets cannot be applied in our context, because we modify renewal processes through inhomogeneous perturbations and conditioning (see (1.4)–(1.9) below). For this reason, in Appendix A we review and develop a general framework to study convergence of random closed sets of R, based on a natural notion of finite-dimensional distributions. 1.2 Disordered pinning models Let ω := (ωn )n∈N be i.i.d. random variables (independent of the renewal process τ ), which represent the disorder. Probability and expectation for ω will be denoted respectively by P and E. We assume that E[ωn ] = 0,

Var(ωn ) = 1,

∃t0 > 0 : (t) := log E[etωn ] < ∞ for |t| ≤ t0 . (1.3) The disordered pinning model is a random probability law PωN ,β,h on subsets of {0, . . . , N }, indexed by realizations ω of the disorder, the system size N ∈ N, the disorder strength β > 0 and bias h ∈ R, defined by the following Gibbs change of measure of the renewal process τ : PωN ,β,h (τ ∩ [0, N ]) P(τ ∩ [0, N ])

:=

1

e Z ωN ,β,h

N

n=1 (βωn −(β)+h)1{n∈τ }

,

(1.4)

where the normalizing constant  N Z ωN ,β,h := E e n=1 (βωn −(β)+h)1{n∈τ }

(1.5)

is called the partition function. In words, we perturb the law of the renewal process τ in the interval [0, N ], by giving rewards/penalties (βωn − (β) + h) to each visited site n ∈ τ . (The presence of the factor (β) in (1.4)–(1.5), which just corresponds to a translation of h, allows to have normalized weights E[eβωn −(β) ] = 1 for h = 0).

123

20

F. Caravenna et al.

The properties of the model PωN ,β,h , especially in the limit N → ∞, have been studied in depth in the recent mathematical literature (see e.g. [13,19,20] for an overview). In this paper we focus on the problem of defining a continuum analogue of PωN ,β,h . Since under the “free law” P the rescaled renewal process τ/N converges in distribution to the α-stable regenerative set τ , it is natural to ask what happens under the “interacting law” PωN ,β,h . Heuristically, in the scaling limit the i.i.d. random variables (ωn )n∈N should be replaced by a one-dimensional white noise (dWt )t∈[0,∞) , where W = (Wt )t∈[0,∞) denotes a standard Brownian motion (independent of τ ). Looking at (1.4), a natural candidate for the scaling limit of τ/N under PωN ,β,h would be the random measure Pα;W T,β,h on C defined by dPα;W T,β,h dPα

(τ ∩ [0, T ]) :=

1 Z α;W T,β,h

T

e

0

1{t∈τ } βdWt + h− 21 β 2 dt

,

(1.6)

where the continuum partition function Z α;W T,β,h would be defined in analogy to (1.5). The problem is that a.e. realization of the α-stable regenerative set τ has zero Lebesgue measure, hence the integral in (1.6) vanishes, yielding the “trivial” definition Pα;W T,β,h = Pα . These difficulties turn out to be substantial and not just technical: as we shall see,   a non-trivial scaling limit of PωN ,β,h does exist, but, for α ∈ 21 , 1 , it is not absolutely continuous with respect to the law Pα [hence no formula like (1.6) can hold]. Note that an analogous phenomenon happens for the directed polymer in random environment [4]. 1.3 Main results We need to formulate an additional assumption on the renewal processes that we consider. Introducing the renewal function u(n) := P(n ∈ τ ) =



P(τk = n),

k=0

assumption (1.1) yields u(n + )/u(n) → 1 as n → ∞, provided  = o(n) (see (2.10) below). We ask that the rate of this convergence is at least a power-law of n : ∃C, n 0 ∈ (0, ∞), ε, δ ∈ (0, 1] :

   δ  u(n + )     ∀n ≥ n 0 , 0 ≤  ≤ εn.  u(n) −1 ≤ C n (1.7)

Remark 1.2 As we discuss in Appendix B, condition (1.7) is a very mild smoothness requirement, that can be verified in most situations. E.g., it was shown by Alexander [2] that for any α > 0 and slowly varying L(·), there exists a Markov chain X on N0 with ±1 steps, called Bessel-like random walk, whose return time to 0, denoted by T , is such that

123

The continuum disordered pinning model

K (n) := P(T = 2n) =

 L(n) n 1+α

21

as n → ∞,

with  L(n) ∼ L(n).

(1.8)

We prove in Appendix B that any such walk always satisfies (1.7). Recall that C denotes the compact Polish space of closed subsets of R. We denote by M1 (C) the space of Borel probability measures on C, which is itself a compact Polish space, equipped with the topology of weak convergence. We will work with a conditioned version of the disordered pinning model (1.4), defined by ω Pω,c N ,β,h ( · ) := P N ,β,h ( · |N ∈ τ ).

(1.9)

ω,c (In order to lighten notation, when N ∈ / N we agree that Pω,c N ,β,h := P N ,β,h ). Recalling (1.2), let us introduce the notation

τ ∩ [0, T ] under Pω,c N T,β N ,h N . N (1.10) (d(τ/N )) is a probability law on For a fixed realization of the disorder ω, Pω,c N T,β N ,h N ω,c C, i.e. an element of M1 (C). Since ω is chosen randomly, the law P N T,β N ,h N (d(τ/N )) is a random element of M1 (C), i.e. a M1 (C)-valued random variable. Our first main is the convergence in distribution of this random variable,   result provided α ∈ 21 , 1 and the coupling constants β = β N and h = h N are rescaled appropriately: Pω,c N T,β N ,h N (d(τ/N )) := law of the rescaled set

β N := βˆ

L(N ) N

α− 21

,

h N := hˆ

L(N ) , Nα

for N ∈ N, βˆ > 0, hˆ ∈ R.

(1.11)

Theorem 1.3 (Existence and universality of the CDPM) Fix α ∈ ( 21 , 1), T > 0, βˆ > 0, hˆ ∈ R. There exists a M1 (C)-valued random variable Pα;W,c , called the ˆ hˆ T,β, (conditioned) continuum disordered pinning model (CDPM), which is a function of ˆ and of a standard Brownian motion W = (Wt )t≥0 , with ˆ h) the parameters (α, T, β, the following property: • for any renewal process τ satisfying (1.1) and (1.7), and β N , h N defined as in (1.11); • for any i.i.d. sequence ω satisfying (1.3); the law Pω,c N T,β N ,h N (d(τ/N )) of the rescaled pinning model (1.10), viewed as a M1 (C)valued random variable, converges in distribution to Pα;W,c as N → ∞. ˆ ˆ T,β,h

We refer to Sect. 1.4 for adiscussion on the universality of the CDPM. We stress that the restriction α ∈ 21 , 1 is substantial and not technical, being linked with the issue of disorder relevance, as we explain in Sect. 1.4 (see also [10]). Let us give a quick explanation of the choice of scalings (1.11). This is the canonical scaling under which the partition function Z ωN ,β N ,h N in (1.5) has a nontrivial continuum limit. To see this, write

123

22

F. Caravenna et al.



Z ωN ,β,h

 N  β,h 1 + εn 1n∈τ =E n=1

=1+

N



k=1 1≤n 1 0,   βˆ > 0, hˆ ∈ R, the averaged law E Pα;W,c of the CDPM is absolutely continuous with ˆ ˆ T,β,h

respect to the reference law Pα;c T . It follows that any typical property of the reference α;c law P T is also a typical property of the quenched law Pα;W,c , for a.e. realization of ˆ hˆ T,β, W: ∀A ⊆ C such that Pα;c T (A) = 1 :

Pα;W,c (A) = 1 for P-a.e. W. ˆ ˆ T,β,h

(1.14)

of the CDPM is In particular, for a.e. realization of W , the quenched law Pα;W,c ˆ hˆ T,β, supported on closed subsets of [0, T ] with Hausdorff dimension α. It is tempting to deduce from (1.14) the absolute continuity of the quenched law Pα;W,c with respect to the reference law Pα;c T , for a.e. realization of W , but this is false. ˆ ˆ T,β,h

  Theorem 1.5 (Singularity of the quenched CDPM) For all α ∈ 21 , 1 , T > 0, βˆ > 0, of the CDPM is singular hˆ ∈ R and for a.e. realization of W , the quenched law Pα;W,c ˆ ˆ with respect to the reference law Pα;c T :

T,β,h

α;W,c for P-a.e. W, ∃A ⊆ C such that Pα;c T (A) = 1 and P ˆ ˆ (A) = 0. T,β,h

(1.15)

The seeming contradiction between (1.14) and (1.15) is resolved noting that in (1.14) one cannot exchange “∀A ⊆ C” and “for P-a.e. W ”, because there are uncountably many A ⊆ C (and, of course, the set A appearing in (1.15) depends on the realization of W ). We conclude our main results with an explicit characterization of the CDPM. As we discuss in Appendix A, each closed subset C ⊆ R can be identified with two non-decreasing and right-continuous functions gt (C) and dt (C), defined for t ∈ R by gt (C) := sup{x : x ∈ C ∩ [−∞, t]},

dt (C) := inf{x : x ∈ C ∩ (t, ∞]}. (1.16)

As a consequence, the law of a random closed subset X ⊆ R is uniquely determined by the finite dimensional distributions of the random functions (gt (X ))t∈R and 2k (dt (X ))t∈R , i.e. by the probability laws on R given, for k ∈ N and −∞ < t1 < t2 < . . . < tk < ∞, by   P gt1 (X ) ∈ dx1 , dt1 (X ) ∈ dy1 , . . . , gtk (X ) ∈ dxk , dtk (X ) ∈ dyk .

(1.17)

As a further simplification, it is enough to focus on the event that X ∩ [ti , ti+1 ] = ∅ for all i = 1, . . . , k, that is, one can restrict (x1 , y1 , . . . , xk , yk ) in (1.17) on the following set:  (k) Rt0 ,...,tk+1 := (x1 , y1 , . . . , xk , yk ) : xi ∈ [ti−1 , ti ], yi ∈ [ti , ti+1 ] for i = 1, . . . , k,  such that yi ≤ xi+1 for i = 1, . . . , k − 1 , (1.18)

123

24

F. Caravenna et al.

with t0 = −∞ and tk+1 := +∞. The measures (1.17) restricted on the set (1.18) will be called restricted finite-dimensional distributions (f.d.d.) of the random set X (see §A.3). We can characterize the CDPM by specifying its restricted f.d.d.. We need two ingredients: (1) The restricted f.d.d. of the α-stable regenerative set conditioned to visit T , i.e. of the reference law Pα;c T in (1.13): by Proposition A.8, these are absolutely continuous with respect to the Lebesgue measure on R2k , with the following density (with y0 := 0): fα;c T ;t1 ,...,tk (x 1 , y1 , . . . , x k , yk ) =

 k



i=1

)1−α

(xi − yi−1

 (yi − xi

)1+α

T 1−α , (T − yk )1−α

(1.19) α sin(πα) with Cα := , π

(1.20)

where we restrict (x1 , y1 , . . . , xk , yk ) on the set (1.18), with t0 = 0 and tk+1 := T . (2) A family of continuum partition functions for our model: 

Z α;W,c (s, t) ˆ ˆ β,h

 0≤s≤t 0, βˆ > 0, hˆ ∈ R and   (s, t) 0≤s≤t 0) has the same qualitative behavior as the homogeneous model (β = 0), provided the disorder is sufficiently weak (β  1); • relevant if, on the other hand, an arbitrarily small amount of disorder (any β > 0) alters the qualitative behavior of the homogeneous model (β = 0). Recalling that α is the exponent appearing in (1.1), it is known that disorder is irrelevant for pinning models when α < 21 and relevant when α > 21 , while the case α = 21 is called marginal and is more delicate (see [20] and the references therein for an overview). It is natural to interpret our results from this perspective. For simplicity, in the sequel we set h N := hˆ L(N )/N α , as in (1.11), and we use the notation Pω,c N T,β N ,h N (d(τ/N )) (1.10), for the law of the rescaled set τ/N under the pinning model. In the homogeneous case (β = 0), it was shown in [31, Theorem 3.1]1 that the α;c weak limit of Pα;c on C which N T,0,h N (d(τ/N )) as N → ∞ is a probability law P ˆ T,0,h

is absolutely continuous with respect to the reference law Pα;c T (recall (1.13)): dPα;c

T,0,hˆ (τ ) dPα;c T

ˆ

=

eh LT (τ ) E[ehˆ LT (τ ) ]

,

(1.23)

where LT (τ ) denotes the so-called local time associated to the regenerative set τ . We stress that this result holds with no restriction on α ∈ (0, 1). 1 Actually [31] considers the non-conditioned case (1.4), but it can be adapted to the conditioned case.

123

26

F. Caravenna et al.

  Turning to the disordered model β > 0, what happens for α ∈ 0, 21 ? In analogy with [9,11], we conjecture that for fixed β > 0 small enough, the limit in distribution of Pω,c N T,β,h N (d(τ/N )) as N → ∞ is the same as for the homogeneous model (β = 0),   i.e. the law Pα;c ˆ defined in (1.23). Thus, for α ∈ 0, 21 , the continuum model is T,0,h non-disordered (deterministic) and absolutely continuous with respect to the reference law.   This is in striking contrast with the case α ∈ 21 , 1 , where our results show that the is truly disordered and singular with respect to the reference continuum model Pα;W,c ˆ hˆ T,β,   law (Theorems 1.3, 1.4, 1.5). In other terms, for α ∈ 21 , 1 , disorder survives in the scaling limit (even though β N , h N → 0) and breaks down the absolute continuity with respect to the reference law, providing a clear manifestation of disorder relevance. We refer to [10] for a general discussion on disorder relevance in our framework. of the CDPM is a random probability law 2. (Universality) The quenched law Pα;W,c ˆ hˆ T,β, on C, i.e. a random variable taking values in M1 (C). Its distribution is a probability law on the space M1 (C)—i.e. an element of M1 (M1 (C))—which is universal: it depends on few macroscopic parameters (the time horizon T , the disorder strength ˆ hˆ and the exponent α) but not on finer details of the discrete model from and bias β, which it arises, such as the distributions of ω1 and of τ1 : all these details disappear in the scaling limit. Another important universal aspect of the CDPM is linked to phase transitions. We do not explore this issue here, referring to [10, §1.3] for a detailed discussion, but we mention that the CDPM leads to sharp predictions about the asymptotic behavior of the free energy and critical curve of discrete pinning models, in the weak disorder regime λ, h → 0. 3. (Bessel processes) In this paper we consider pinning models built on top of general renewal processeses τ = (τk )k∈N0 satisfying (1.1) and (1.7). In the special case when the renewal process is the zero level set of a Bessel-like random walk [1] (recall Remark 1.2), one can define the pinning model (1.4), (1.9) as a probability law on random walk paths (and not only on their zero level set). Rescaling the paths diffusively, one has an analogue of Theorem 1.3, in which the CDPM is built as a random probability law on the space C([0, T ], R) of continuous functions from [0, T ] to R. Such an extended CDPM is a continuous process (X t )t∈[0,T ] , that can be heuristically described as a Bessel process of dimension δ = 2(1 − α) interacting with an independent Brownian environment W each time X t = 0. The “original” CDPM of our Theorem 1.3 corresponds to the zero level set τ := {t ∈ [0, T ] : X t = 0}. We stress that, starting from the zero level set τ , one can reconstruct the whole process (X t )t∈[0,T ] by pasting independent Bessel excusions on top of τ (more precisely, since the open set [0, T ]\τ is a countable union of disjoint open intervals, one attaches a Bessel excursion to each of these intervals).2 This provides a rigorous 2 Alternatively, one can write down explicitly the f.d.d. of (X ) t t∈[0,T ] in terms of the continuum partition functions Z α;W,c (s, t) (see Sect. 2). We skip the details for the sake of brevity. ˆ hˆ β,

123

The continuum disordered pinning model

27

definition of (X t )t∈[0,T ] in terms of τ and shows that the zero level set is indeed the fundamental object. 4. (Infinite-volume limit) Our continuum model Pα;W,c is built on a finite interval ˆ ˆ T,β,h

[0, T ]. An interesting open problem is to let T → ∞, proving that Pα;W,c converges ˆ ˆ Pα;W,c . ˆ hˆ ∞,β,

T,β,h

in distribution to an infinite-volume CDPM Such a limit law would inherit scaling properties from the continuum partition functions, Theorem 2.4 (iii). (See also [29] for related work in the non-disordered case βˆ = 0). 1.5 Organization of the paper The rest of the paper is organized as follows. • In Sect. 2, we study the properties of continuum partition functions. • In Sect. 3, we prove Theorem 1.6 on the characterization of the CDPM, which also yields Theorem 1.3. • In Sect. 4, we prove Theorems 1.4 and 1.5 on the relations between the CDPM and the α-stable regenerative set. • In Appendix A, we describe the measure-theoretic background needed to study random closed subsets of R, which is of independent interest. • Lastly, in Appendices B and C we prove some auxiliary estimates. 2 Continuum partition functions as a process   (s, t) 0≤s≤t 0, hˆ ∈ R. Let τ be a renewal process satisfying (1.1) and (1.7), and ω be an i.i.d. sequence satisfying (1.3). For every N ∈ N, define β N , h N by (recall (1.11)) ⎧ L(N ) ⎪ ⎪ ⎨β N := βˆ 1 N α− 2 ⎪ L(N ) ⎪ ⎩h N := hˆ Nα

⎧ βˆ ⎪ ⎪ ⎨β N := √ 1  N for α > 1. (2.2) for α ∈ 2 , 1 , ⎪ ˆ h ⎪ ⎩h := N N  ω,c  As N → ∞ the two-parameter family Z β N ,h N (s N , t N ) 0≤s≤t pq − d, uniformly in N ∈ N, x, y ∈ [0, 1]d . Then E[B( f N )] is bounded uniformly in N , hence {B( f N )} N ∈N is tight. If the functions f N are equibounded at some point (e.g. f N (0) = 1 for every N ∈ N), the tightness of B( f N ) entails the tightness of { f N } N ∈N , by the Arzelà-Ascoli theorem [6, Theorem 7.3]. (s N , t N ))0≤s≤t≤1 } N ∈N , it then suffices to show To prove the tightness of {(Z βω,c N ,h N that  p # η (s1 N , t1 N ) − Z βω,c (s2 N , t2 N ) ≤ C (s1 − s2 )2 + (t1 − t2 )2 , E  Z βω,c N ,h N N ,h N (2.12) which by triangle inequality, translation invariance and symmetry can be reduced to ∃C > 0, p ≥ 1, η > 2 :

123

 p ω,c  ≤ C|t − s|η , (0, t N ) − Z (0, s N ) E  Z βω,c β N ,h N N ,h N (2.13)

The continuum disordered pinning model

31

uniformly in N ∈ N and 0 ≤ s < t ≤ 1. (Conditions pq > 2d and η > pq −d are then ω,c fulfilled by any q ∈ ( 4p , 2+η p ), since d = 2). Since Z β N ,h N (0, ·) is defined on [0, ∞) via linear interpolation, it suffices to prove (2.13) for s, t with s N , t N ∈ {0} ∪ N. Step 2. Polynomial chaos expansion To simplify notation, let us denote

N ,r :=

Z βω,c (0, r ) N ,h N

=E

r −1 

e

(β N ωn −(β N )+h N )1{n∈τ }

n=1

   r ∈ τ 

for r ∈ N,

and N ,0 := 1. Since e x1{n∈τ } = 1 + (e x − 1)1{n∈τ } for all x ∈ R, we set ξ N ,i := eβ N ωi −(β N )+h N − 1,

(2.14)

and rewrite N ,r as a polynomial chaos expansion:

N ,r

r −1   1 + ξ N ,i 1{i∈τ } =E

   r ∈ τ = 

i=1

P(I ⊂ τ |r ∈ τ )

I ⊂{1,...,r −1}



ξ N ,i ,

i∈I

(2.15)

$ using the notation {I ⊂ τ } := i∈I {i ∈ τ }. Recalling (2.2) and (1.3), it is easy to check that

E[ξ N ,i ] = eh N − 1 = h N + O(h 2N ), % #   % Var(ξ N ,i ) = e2h N e(2β N )−2(β N ) − 1 = β N2 + O(β N3 ) = β N + O(β N2 ), (2.16) where we used the fact that h N = o(β N ) and we Taylor expanded (t) := log E[etω1 ], noting that (0) =  (0) = 0 and  (0) = 1. Thus h N and β N are approximately the mean and standard deviation of ξ N ,i . Let us rewrite N ,r in (2.15) using normalized variables ζ N ,i :

N ,r =

I ⊂{1,...,r −1}

ψ N ,r (I )



ζ N ,i ,

where

ζ N ,i :=

i∈I

1 ξ N ,i , βN

(2.17)

where ψ N ,r (∅) := 1 and for I = {n 1 < n 2 < · · · < n k } ⊂ N, recalling (2.10), we can write |I |

ψ N ,r (I ) = ψ N ,r (n 1 , . . . , n k ) := β N P(I ⊂ τ |r ∈ τ ) = (β N )k

k+1 1  u(n i − n i−1 ), u(r ) i=1 (2.18)

with n 0 := 0, n k+1 := r . (0, s N ) = N ,q and Z βω,c (0, t N ) = N ,r , To prove (2.13), we write Z βω,c N ,h N N ,h N with q := s N and r := t N , so that 0 ≤ q < r ≤ N . For a given truncation level

123

32

F. Caravenna et al.

m = m(q, r, N ) ∈ (0, q), that we will later choose as & √ 0 if q ≤ N (r − q) m = m(q, r, N ) := √ q − N (r − q) otherwise,

(2.19)

so that 0 ≤ m < q < r ≤ N , we write

N ,r − N ,q = 1 + 2 − 3 with

1 =

  ψ N ,r (I ) − ψ N ,q (I ) ζ N ,i ,

I ⊂{1,...,m}

2 =

ψ N ,r (I )

I ⊂{1,...,r −1} I ∩{m+1,...,r −1}=∅

3 =



i∈I

ζ N ,i ,

and

i∈I

ψ N ,q (I )

I ⊂{1,...,q−1} I ∩{m+1,...,q−1}=∅



ζ N ,i .

(2.20)

i∈I

To establish (2.13) and hence tightness, it suffices to show that for each i = 1, 2, 3, E[|i | p ] ≤ C

∃C > 0, p ≥ 1, η > 2 :

r − q η ∀N ∈ N, 0 ≤ q < r ≤ N . N (2.21)

Step 3. Change of measure We now estimate the moments of ξ N ,i defined in (2.14). Since (a + b)2k ≤ 22k−1 (a 2k + b2k ), for all k ∈ N, and h N = O(β N2 ) by (2.2), we can write  2k  + 22k−1 (e−(β N )+h N − 1)2k E[ξ N2k,i ] ≤ 22k−1 e2k(h N −(β N )) E eβ N ωi − 1  2k   βN 1 2k tωi ≤ C(k) β N E ωi e dt + O(β N4k + h 2k N) βN 0  βN ≤ C(k)β N2k−1 E[ωi2k e2ktωi ]dt + o(β N2k ) = O(β N2k ), (2.22) 0

because E[ωi2k e2ktωi ] is uniformly bounded for t ∈ [0, t0 /4k] by our assumption (1.3). Recalling (2.16), (2.17) and (2.2), the random variables (ζ N ,i )i∈N are i.i.d. with E[ζ N ,i ]



N →∞

hˆ 1 √ , βˆ N

Var[ζ N ,i ]



N →∞

1,

sup E[(ζ N ,i )2k ] < ∞. (2.23)

N ,i∈N

It follows, in particular, that {ζ N2 ,i }i,N ∈N are uniformly integrable. We can then apply a change of measure result established in [10, Lemma B.1], which asserts that we

123

The continuum disordered pinning model

33

can construct i.i.d. random variables ( ζ N ,i )i∈N with marginal distribution P( ζ N ,i ∈ dx) = f N (x)P(ζ N ,i ∈ dx), for which there exists C > 0 such that for all p ∈ R and i, N ∈ N E[ ζ N ,i ] = 0,

√ E[ ζ N2 ,i ] ≤ 1 + C/ N ,

E[ f N (ζ N ,i ) p ] ≤ 1 + C/N . (2.24) i be the analogue of i constructed from the  ζ N ,i ’s instead of the ζ N ,i ’s. By Let  Hölder, and

  N N    l−1  l−1 l−1 − l−1 l l E |i | f N (ζ N ,i ) f N (ζ N ,i ) = E |i | i=1

i=1

 l−1  N  l−1  C  i |l l . i |l l E f N (ζ N ,1 )1−l l ≤ e l E | ≤ E | (·, ·)} N ∈N , is thus reduced to showRelation (2.21), and hence the tightness of {Z βω,c N ,h N ing r − q η   i |l ≤ C for all N ∈ N and 0 ≤ q < r ≤ N , (2.25) E | N l for some l ∈ N, l ≥ 2 and η > 0 satisfying η > 2 l−1 .

3 |l ] is exactly the same as 2 |l ]. We note that the bound for E[| Step 4. Bounding E[| l 2 as 2 | ], and hence will be omitted. First we write  that for E[| 2 = 

r −1

(k) ,  2

(k) :=  2

where

|I |=k,I ⊂{1,...,r −1} I ∩{m+1,...,r −1}=∅

k=1

ψ N ,r (I )



 ζ N ,i ,

(2.26)

i∈I

(k) consisting of all terms of degree k. The hypercontractivity established in with  2 [26, Prop. 3.16 & 3.12] allows to estimates moments of order l in terms of moments of order 2: more precisely, setting X  p := E[|X | p ]1/ p , we have for all l ≥ 2 2 ll := E[| 2 |l ] ≤ 

'r −1

(l (k) l  2

'r −1 (l

(k)  2 , ≤ (cl )k  2

k=1

(2.27)

k=1

 √ ζ  where cl := 2 l − 1 max N ∈N ζ N ,1 l is finite and depends only on l, by (2.23). N ,1 2

(k) 2 . Let us recall the definition of ψ N ,r in We now turn to the estimation of  2 √ (2.18). It follows by (2.24) that Var( ζ N ,1 ) ≤ 1 + C/ N ≤ 2 for all N large. We then have k−1  (k) 2   (k) 22 = E  =  2 2 y=0

1≤n 1 k) +

1

1 + (n + )2(η−1)

(

n 2(η−1)   1∧(2η−2) 1 ≤ C  α  + 2(η−1) ≤ C  . n n n k=n+1

This establishes (4.27) with δ = 1 ∧ (2η − 2).

 

Lemma B.2 Let τ be a non-terminating renewal process satisfying (1.1), with α ∈ (0, 1), such that 2τ1 has the same distribution as the first return to zero of a nearestneighbor Markov chain on N0 with ±1 increments (Remark 1.2). Then (4.27) holds true. Proof Let X and Y be two copies of such a Markov chain, starting at the origin at times −2 and 0 respectively, so that u(n + ) = P(X 2n = 0)

and

u(n) = P(Y2n = 0).

(Although X n is defined for n ≥ −2, we only look at it for n ≥ 0.) We can couple X and Y such that they are independent until they meet, at which time they coalesce. Since X and Y are nearest neighbor walks, X 2n = 0 implies Y2n = 0, and 0 ≤ P(Y2n = 0) − P(X 2n = 0) = u(n) − u(n + ) = P(Y2n = 0 = X 2n ) = P(Y2n = 0, X t = 0 ∀ t ∈ [0, 2n]) ≤ P(Y2n = 0)P(X t = 0 ∀ t ∈ [0, 2n]) ≤ u(n)

−1

k=0

≤ u(n)P(τ1 > n)

−1

k=0

123

u(k),

u(k)P(τ1 > n +  − k)

The continuum disordered pinning model

57

where by the properties of regularly varying functions, see [8, Prop. 1.5.8 and 1.5.10], P(τ1 > n) =



L(k) L(n) ∼ 1+α k αn α

as n → ∞,

k=n+1 −1

u(k) =

k=0

−1

Cα (1 + o(1)) k=0

L(k)k 1−α



Cα α as  → ∞. αL()

(4.28)

Observe that, for every ε > 0 there exists n 0 < ∞ such that L(n)/L() ≤ (n/)ε for n ≥ n 0 and  ≤ n4 , by Potter bounds [8, Theorem 1.5.6], hence L(n) u(n) − u(n + ) ≤C u(n) L()



 α  α−ε   ≤C , n n

and therefore (4.27) holds true for any δ < α.

 

Appendix C: An integral estimate Lemma C.1 Let χ ∈ [0, 1). Then there exist C1 , C2 > 0 such that for all k ∈ N, 

 ···

0

The continuum disordered pinning model.

Any renewal processes on [Formula: see text] with a polynomial tail, with exponent [Formula: see text], has a non-trivial scaling limit, known as the ...
831KB Sizes 0 Downloads 8 Views