getdoc5373. 393KB Jun 04 2011 12:04:23 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b il i t y Vol. 16 (2011), Paper no. 20, pages 552–586.

Journal URL

http://www.math.washington.edu/~ejpecp/

Collision local time of transient random walks and

intermediate phases in interacting stochastic systems

∗

Matthias Birkner

1

_{, Andreas Greven}

2

_{, Frank den Hollander}

3 4

Abstract

In a companion paper[6], a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words ac-cording to a renewal process. We apply this LDP to prove that the radius of convergence of the generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk onZd_{, d≥ 1, both starting from the origin, strictly increases}

when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds when the random walk is transient but not strongly transient. The presence of these gaps implies the existence of anintermediate phasefor the long-time be-haviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments .

Key words: Random walks, collision local time, annealed vs. quenched, large deviation princi-ple, interacting stochastic systems, intermediate phase.

AMS 2000 Subject Classification:Primary 60G50, 60F10, 60K35, 82D60.

Submitted to EJP on December 24, 2008, resubmitted upon invitation to EJP on March 27, 2010, final version accepted January 11, 2011.

1_{Institut für Mathematik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany.}

Email:birkner@mathematik.uni-mainz.de

2_{Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1}1

2, 91054 Erlangen, Germany.

Email:greven@mi.uni-erlangen.de

3_{Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands.}

Email:denholla@math.leidenuniv.nl

4_{EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands}

∗_{This work was supported in part by DFG and NWO through the Dutch-German Bilateral Research Group “Mathematics}

of Random Spatial Models from Physics and Biology”. MB and AG are grateful for hospitality at EURANDOM. We thank an anonymous referee for careful reading and helpful comments.

1

Introduction and main results

In this paper, we derivevariational representations for the radius of convergence of the generating

functions of the collision local time of two independent copies of a symmetric and transient random walk, both starting at the origin and running in discrete or in continuous time, when the average is taken w.r.t. one, respectively, two random walks. These variational representations are subsequently used to establish the existence of an intermediate phase for the long-time behaviour of a class of interacting stochastic systems.

1.1

Collision local time of random walks

1.1.1 Discrete time

LetS= (S_k)∞_k=0 andS′= (S′_k)∞_k=0 be two independent random walks onZd_{, d≥1, both starting at}

the origin, with anirreducible,symmetricandtransienttransition kernelp(_·,_·). Writepnfor then-th convolution power ofp, and abbreviatepn(x):=pn(0,x), x ∈Zd. Suppose that

lim

n→∞

logp2n(0)

logn =:−α, α∈[1,∞). (1.1)

WriteP_{to denote the joint law ofS,S}′_{. Let}

V =V(S,S′):=

∞ X

k=1

1_{S

k=S′k} (1.2)

be thecollision local timeofS,S′, which satisfiesP_{(V <∞) =1 by transience, and define} z1 := sup

¦

z≥1: E”_zV _|S—_{<∞ S-a.s.}©_{, (1.3)}

z₂ := sup¦z_≥1: E”zV—<∞©. (1.4)

The lower indices indicate the number of random walks being averaged over. Note that, by the tail triviality ofS, the range ofz’s for whichE[zV _|S]converges isS-a.s. constant.1

Let E := Zd, let Ee = _∪n∈NEn be the set of finite words drawn from E, and let Pinv(Ee N

) denote

the shift-invariant probability measures on EeN, the set of infinite sentences drawn from Ee. Define

f: Ee_→[0,_∞)via

f((x1, . . . ,xn)) =

pn(x₁+_{· · ·}+x_n)

p2⌊n/2⌋(0) [2 ¯G(0)−1], n∈N, x1, . . . ,xn∈E, (1.5) where ¯G(0) =P∞_n=0p2n(0)is the Green function at the origin associated with p2(_·,_·), which is the transition matrix ofS−S′, and p2⌊n/2⌋(0)>0 for alln∈Nby the symmetry ofp(_·,_·). The following variational representations hold forz1andz2.

1_{Note thatP(V=}

Theorem 1.1. Assume(1.1). Then z₁=1+exp[₋r₁], z₂=1+exp[₋r₂]with

r1 ≤ sup

Q∈Pinv_(eEN )

¨Z e

E

(π1Q)(d y)logf(y)−Ique(Q)

«

, (1.6)

r2 = sup

Q∈Pinv_(eEN₎

¨Z e

E

(π1Q)(d y)logf(y)−Iann(Q)

«

, (1.7)

where π1Q is the projection of Q onto E, while Ie que and Iann are the rate functions in the quenched, respectively, annealed large deviation principle that is given in Theorem2.2, respectively,2.1below with (see(2.4),(2.7)and(2.13–2.14))

E=Zd_{, ν}(x) =p(x), x∈E, ρ(n) =p2⌊n/2⌋(0)/[2 ¯G(0)₋1], n∈N_{. (1.8)}

Let

Perg,fin(EeN) =_{Q∈ Pinv(eEN): Qis shift-ergodic,mQ<∞}, (1.9)

where mQ is the average word length underQ, i.e., mQ =

R e

E(π1Q)(y)τ(y)with τ(y)the length

of the word y. Theorem 1.1 can be improved under additional assumptions on the random walk,

namely,2

P

x∈Zd

kx_kδp(x)<∞ for someδ >0, (1.10)

lim inf

n→∞

log[pn_(S

n)/p2⌊n/2⌋(0) ]

logn ≥0 S−a.s., (1.11)

inf

n∈N

Elog[pn(S_n)/p2⌊n/2⌋(0) ]>−∞. (1.12)

Theorem 1.2. Assume(1.1)and(1.10–1.12). Then equality holds in(1.6), and

r1 = sup

Q∈Perg,fin_(eEN )

¨Z e

E

(π1Q)(d y)logf(y)−Ique(Q)

«

∈R_{, (1.13)}

r2 = sup

Q∈Perg,fin_(eEN )

¨Z e

E

(π1Q)(d y)logf(y)−Iann(Q)

«

∈R. (1.14)

In Section 6 we will exhibit classes of random walks for which (1.10–1.12) hold. We believe that (1.11–1.12) actually hold in great generality.

BecauseIque_≥Iann, we haver_1≤r₂, and hencez_2≤z₁(as is also obvious from the definitions ofz₁

andz₂). We prove that strict inequality holds under the stronger assumption that p(_·,_·)isstrongly transient, i.e.,P∞_n=1npn(0)<∞. This excludesα∈(1, 2)and part ofα=2 in (1.1).

Theorem 1.3. Assume(1.1). If p(_·,_·)is strongly transient, then1<z2<z1≤ ∞. 2_{By the symmetry of p(}

·,_·), we have supx∈Zdp

n_(x)

≤ p2⌊n/2⌋_{(0) (see (3.15)), which implies that sup}

n∈Nsupx∈Zd

SinceP(V = k) = (1₋F¯)F¯k, k_∈N_{∪ {}0_}, with ¯F :=P _∃k_∈N: S_k = S_k′, an easy computation givesz2=1/F¯. But ¯F=1−[1/G¯(0)](see Spitzer[27], Section 1), and hence

z₂=G¯(0)/[G¯(0)₋1]. (1.15)

Unlike (1.15), no closed form expression is known for z1. By evaluating the function inside the

supremum in (1.13) at a well-chosenQ, we obtain the following upper bound.

Theorem 1.4. Assume(1.1)and(1.10–1.12). Then

z1≤1+

X

n∈N e−h(pn)

!₋1

<∞, (1.16)

where h(pn) =₋P_x∈Zdpn(x)logpn(x)is the entropy of pn(·).

There are symmetric transient random walks for which (1.1) holds withα=1. Examples are any

transient random walk onZ_{in the domain of attraction of the symmetric stable law of index 1 on}

R_{, or any transient random walk on}Z2_{in the domain of (non-normal) attraction of the normal law}

onR2. If in this situation (1.10–1.12) hold, then the two threshold values in (1.3–1.4) agree.

Theorem 1.5. If p(_·,_·)satisfies(1.1)withα=1and(1.10–1.12), then z₁=z₂.

1.1.2 Continuous time

Next, we turn the discrete-time random walks S and S′ into continuous-time random walks eS =

(S_t)_t≥0 andSe′_{= (}e_S′

t)t≥0 by allowing them to make steps at rate 1, keeping the same p(·,·). Then

the collision local time becomes

e

V :=

Z _∞

0

1_{eS

t=eS′t}d t. (1.17)

For the analogous quantitiesez1 andez2, we have the following.3

Theorem 1.6. Assume(1.1). If p(_·,·)is strongly transient, then1<ez2<ez1≤ ∞.

An easy computation gives

log_ez₂=2/G(0), (1.18)

whereG(0) =P∞_n=0pn(0)is the Green function at the origin associated with p(_·,_·). There is again no simple expression forez1.

Remark 1.7. An upper bound similar to(1.16)holds forze1as well. It is straightforward to show that z1<∞andez1<∞as soon as p(·)has finite entropy.

3_{For a symmetric and recurrent random walk again triviallyze}

1.1.3 Discussion

Our proofs of Theorems 1.3–1.6 will be based on the variational representations in Theorem 1.1–1.2. Additional technical difficulties arise in the situation where the maximiser in (1.7) has infinite mean

word length, which happens precisely when p(_·,_·)is transient but not strongly transient. Random

walks with zero mean and finite variance are transient for d ≥ 3 and strongly transient for d ≥5

(Spitzer[27], Section 1).

Conjecture 1.8. The gaps in Theorems1.3 and 1.6are present also when p(_·,·) is transient but not strongly transient providedα >1.

In a 2008 preprint by the authors (arXiv:0807.2611v1), the results in[6]and the present paper were

announced, including Conjecture 1.8. Since then, partial progresshas been made towards settling

this conjecture. In Birkner and Sun[7], the gap in Theorem 1.3 is proved for simple random walk

onZd,d_≥4, and it is argued that the proof is in principle extendable to a symmetric random walk

with finite variance. In Birkner and Sun [8], the gap in Theorem 1.6 is proved for a symmetric

random walk onZ3 _{with finite variance in continuous time, while in Berger and Toninelli [1]the}

gap in Theorem 1.3 is proved for a symmetric random walk onZ3 in discrete time under a fourth

moment condition.

The role of the variational representation for r2 is not to identify its value, which is achieved in

(1.15), but rather to allow for acomparisonwithr1, for which no explicit expression is available. It

is an open problem to prove (1.11–1.12) under mild regularity conditions onS. Note that the gaps

in Theorems 1.3–1.6 do not require (1.10–1.12).

1.2

The gaps settle three conjectures

In this section we use Theorems 1.3 and 1.6 to prove the existence of an intermediate phase for

three classes of interacting particle systems where the interaction is controlled by a symmetric and

transient random walk transition kernel. 4

1.2.1 Coupled branching processes

A.Theorem 1.6provesa conjecture put forward in Greven[17],[18]. Consider a spatial population

model, defined as the Markov process (ηt)t≥0, with η(t) = {ηx(t): x ∈ Zd} where ηx(t) is the

number of individuals at site x at time t, evolving as follows:

(1) Each individual migrates at rate 1 according toa(_·,_·).

(2) Each individual gives birth to a new individual at the same site at rateb.

(3) Each individual dies at rate(1₋p)b.

(4) All individuals at the same site die simultaneously at ratep b.

4_{In each of these systems the case of a symmetric and recurrent random walk is trivial and no intermediate phase is}

Here,a(_·,_·)is an irreducible random walk transition kernel onZd_×Zd, b_∈(0,_∞)is a birth-death

rate, p∈[0, 1]is a coupling parameter, while (1)–(4) occur independently at every x ∈Zd. The

case p = 0 corresponds to a critical branching random walk, for which the average number of

individuals per site is preserved. The case p>0 is challenging because the individuals descending

from different ancestors are no longer independent.

A critical branching random walk satisfies the followingdichotomy(where for simplicity we restrict

to the case wherea(_·,_·)is symmetric): if the initial configurationη0 is drawn from a shift-invariant

and shift-ergodic probability distribution with a positive and finite mean, thenη_t as t → ∞locally

dies out (“extinction”) when a(_·,_·) is recurrent, but converges to a non-trivial equilibrium (“sur-vival”) whena(_·,_·)istransient, both irrespective of the value ofb. In the latter case, the equilibrium has the same mean as the initial distribution and has all moments finite.

For the coupled branching process with p > 0 there is a dichotomy too, but it is controlled by a

subtle interplayofa(_·,·), b andp: extinction holds when a(_·,·)isrecurrent, but also whena(_·,·) is

transientandpissufficiently large. Indeed, it is shown in Greven[18]that ifa(_·,_·)is transient, then there is a uniquep_∗∈(0, 1]such that survival holds forp<p_∗and extinction holds forp>p_∗. Recall the critical values_ez₁,_ez₂introduced in Section 1.1.2. Then survival holds ifE(exp[bpVe]_|eS)<

∞Se-a.s., i.e., ifp<p1 with

p1=1∧(b−1logez1). (1.19)

This can be shown by a size-biasing of the population in the spirit of Kallenberg[23]. On the other

hand, survival with afinite second momentholds if and only ifE(exp[bpVe])<∞, i.e., if and only if

p<p2with

p₂=1_∧(b−1logez₂). (1.20)

Clearly, p_∗≥p_1≥p₂. Theorem 1.6 shows that ifa(_·,_·)satisfies (1.1) and is strongly transient, then

p1>p2, implying that there is an intermediate phase of survival with aninfinite second moment.

B. Theorem 1.3 corrects an error in Birkner [3], Theorem 6. Here, a system of individuals living

on Zd _{is considered subject to migration and branching. Each individual independently migrates}

at rate 1 according to a transient random walk transition kernela(_·,_·), and branches at a rate that depends on the number of individuals present at the same location. It is argued that this system has an intermediate phase in which the numbers of individuals at different sites tend to an equilibrium with a finite first moment but an infinite second moment. The proof was, however, based on a

wrong rate function. The rate function claimed in Birkner[3], Theorem 6, must be replaced by that

in[6], Corollary 1.5, after which the intermediate phase persists, at least in the case where a(_·,·) satisfies (1.1) and is strongly transient. This also affects[3], Theorem 5, which uses[3], Theorem

6, to computez₁in Section 1.1 and finds an incorrect formula. Theorem 1.4 shows that this formula

actually is an upper bound forz1.

1.2.2 Interacting diffusions

Theorem 1.6 proves a conjecture put forward in Greven and den Hollander [19]. Consider the

system X = (X(t))_t≥0, with X(t) = _{X_x(t): x _∈ Zd_{}, of interacting diffusions taking values in}

[0,_∞)defined by the following collection of coupled stochastic differential equations:

d X_x(t) = X

y∈Zd

a(x,y)[X_y(t)₋X_x(t)]d t+pbX_x(t)2 _dW

Here, a(_·,_·)is an irreducible random walk transition kernel on Zd_×Zd, b _∈(0,_∞)is a diffusion constant, and (W(t))_t≥0 withW(t) = _{Wx(t): x ∈ Zd} is a collection of independent standard

Brownian motions onR_{. The initial condition is chosen such thatX}(0)is a invariant and

shift-ergodic random field with a positive and finite mean (the evolution preserves the mean). Note that,

even thoughX a.s. has non-negative paths when starting from a non-negative initial conditionX(0),

we prefer to writeXx(t)as

p

Xx(t)2 in order to highlight the fact that the system in (1.21) belongs

to a more general family of interacting diffusions with Hölder-1

2 diffusion coefficients, see e.g.[19],

Section 1, for a discussion and references.

It was shown in[19], Theorems 1.4–1.6, that ifa(_·,·)issymmetricandtransient, then there exist 0< b_2≤b_∗such that the system in (1.21) locally dies out when b>b_∗, but converges to an equilibrium when 0< b< b_∗, and this equilibrium has afinite second moment when 0< b< b₂ and aninfinite second momentwhen b2 ≤ b < b∗. It was shown in[19], Lemma 4.6, that b∗≥ b∗∗=logez1, and

it was conjectured in[19], Conjecture 1.8, that b_∗> b₂. Thus, as explained in[19], Section 4.2, if

a(_·,_·)satisfies (1.1) and is strongly transient, then this conjecture is correct with

b_∗≥logez₁ > b₂=logez₂. (1.22)

Analogously, by Theorem 1.1 in[8]and by Theorem 1.2 in[7], the conjecture is settled for a class

of random walks in dimensionsd=3, 4 including symmetric simple random walk (which ind=3, 4

is transient but not strongly transient).

1.2.3 Directed polymers in random environments

Theorem 1.3disprovesa conjecture put forward in Monthus and Garel[25]. Leta(_·,_·)be a symmet-ric and irreducible random walk transition kernel onZd_×Zd_{, letS}= (Sk)∞_k=0 be the corresponding

random walk, and let ξ = _{ξ(x,n): x ∈ Zd_{,n ∈} N_{} be i.i.d.} R_{-valued non-degenerate random}

variables satisfying

λ(β):=logE exp[βξ(x,n)]_∈R _∀β∈R. (1.23)

Put

en(ξ,S):=exp

 Xn

k=1

βξ(Sk,k)−λ(β)



, (1.24)

and set

Z_n(ξ):=E_{[en(ξ,S)] =} X

s1,...,sn∈Zd

 

n

Y

k=1

p(s_k−1,s_k)



 e_n(ξ,s), s= (s_k)∞_k=0,s₀=0, (1.25)

i.e., Z_n(ξ)is the normalising constant in the probability distribution of the random walkS whose

paths are reweighted by en(ξ,S), which is referred to as the “polymer measure”. The ξ(x,n)’s

describe a random space-time medium with whichS is interacting, withβ playing the role of the

interaction strength or inverse temperature.

It is well known thatZ= (Z_n)_n∈Nis a non-negative martingale with respect to the family of

sigma-algebras_Fn:=σ(ξ(x,k),x _∈Zd, 1_≤k_≤n),n_∈N. Hence lim

with the event _{Z_∞ = 0_} being ξ-trivial. One speaks of weak disorder if Z_∞ > 0 ξ-a.s. and of

strong disorder otherwise. As shown in Comets and Yoshida[12], there is a unique critical value

β_∗∈[0,_∞] such that weak disorder holds for 0≤ β < β_∗ and strong disorder holds for β > β_∗. Moreover, in the weak disorder region the paths have a Gaussian scaling limit under the polymer measure, while this is not the case in the strong disorder region. In the strong disorder region the paths are confined to a narrow space-time tube.

Recall the critical valuesz1,z2 defined in Section 1.1. Bolthausen[9]observed that

E”_Z2

n

—

=E

h

exp{λ(2β)₋2λ(β)_}Vn

i

, withVn:=

n

X

k=1

1_{Sk=S′k}, (1.27)

whereSandS′are two independent random walks with transition kernelp(_·,_·), and concluded that

ZisL2-bounded if and only ifβ < β2 withβ2∈(0,∞]the unique solution of

λ(2β2)−2λ(β2) =logz2. (1.28)

SinceP(Z_∞>0)_≥E[Z_∞]2/E[Z_∞2]andE[Z_∞] =Z₀=1 for an L2-bounded martingale, it follows that β < β2 implies weak disorder, i.e.,β∗≥ β2. By a stochastic representation of the size-biased

law ofZ_n, it was shown in Birkner[4], Proposition 1, that in fact weak disorder holds ifβ < β1 with β1∈(0,∞]the unique solution of

λ(2β₁)₋2λ(β₁) =logz₁, (1.29)

i.e.,β_∗≥β1. Sinceβ7→λ(2β)−2λ(β)is strictly increasing for any non-trivial law for the disorder

satisfying (1.23), it follows from (1.28–1.29) and Theorem 1.3 that β1 > β2 when a(·,·) satisfies

(1.1) and is strongly transient and when ξ is such that β2 < ∞. In that case the weak disorder

region contains a subregion for whichZ isnot L2-bounded. This disproves a conjecture of Monthus

and Garel[25], who argued thatβ2=β∗.

Camanes and Carmona[10]consider the same problem for simple random walk and specific choices

of disorder. With the help of fractional moment estimates of Evans and Derrida[16], combined with

numerical computation, they show thatβ_∗> β2for Gaussian disorder ind≥5, for Binomial disorder

with small mean ind≥4, and for Poisson disorder with small mean ind≥3.

See den Hollander[21], Chapter 12, for an overview.

Outline

Theorems 1.1, 1.3 and 1.6 are proved in Section 3. The proofs need only assumption (1.1). Theo-rem 1.2 is proved in Section 4, TheoTheo-rems 1.4 and 1.5 in Section 5. The proofs need both assumptions (1.1) and (1.10–1.12)

In Section 2 we recall the LDP’s in [6], which are needed for the proof of Theorems 1.1–1.2 and

their counterparts for continuous-time random walk. This section recalls the minimum from [6]

that is needed for the present paper. Only in Section 4 will we need some of the techniques that were used in[6].

2

Word sequences and annealed and quenched LDP

Notation. We recall the problem setting in [6]. Let E be a finite or countable set of letters. Let

e

E = _∪n∈NEn be the set of finite words drawn from E. Both E and Ee are Polish spaces under the

discrete topology. Let _P(EN) and _P(EeN) denote the set of probability measures on sequences

drawn fromE, respectively,Ee, equipped with the topology of weak convergence. Writeθ andθefor

the left-shift acting onEN, respectively,EeN. WritePinv_(EN

),_Perg_(EN

)andPinv_(EeN

),_Perg_(EeN

)for the set of probability measures that are invariant and ergodic underθ, respectively,θe.

Forν ∈ P(E), letX = (X_i)_i∈Nbe i.i.d. with lawν. Forρ∈ P(N), letτ= (τ_i)_i∈Nbe i.i.d. with law

ρhaving infinite support and satisfying thealgebraic tail property

lim

n→∞

ρ(n)>0

logρ(n)

logn =:−α, α∈[1,∞). (2.1)

(No regularity assumption is imposed on supp(ρ).) Assume thatX andτare independent and write

P_{to denote their joint law. Cut words out ofX according toτ, i.e., put (see Fig. 1)}

T0:=0 and Ti:=Ti−1+τi, i∈N, (2.2)

and let

Y(i):= X_Ti

−1+1,XTi−1+2, . . . ,XTi

, i∈N_{. (2.3)}

Then, under the lawP_{, Y} = (Y(i))_i∈N is an i.i.d. sequence of words with marginal lawq_ρ,ν on eE

given by

q_ρ,ν (x1, . . . ,xn)

:=P _Y(1)= (x1, . . . ,xn)

=ρ(n)ν(x1)· · ·ν(xn), n∈N, x1, . . . ,xn∈E.

(2.4)

τ1

τ2 τ3

τ4

τ5

T₁ T₂ T₃ T₄ T₅ Y(1) Y(2) Y(3) Y(4) Y(5) X

Figure 1: Cutting words from a letter sequence according to a renewal process.

Annealed LDP.ForN _∈N, let(Y(1), . . . ,Y(N))per be the periodic extension of (Y(1), . . . ,Y(N))to an element ofEeN, and define

R_N := 1

N

N_X−1

i=0

δ_θei_(Y(1)_,...,Y(N)₎per ∈ Pinv(Ee N

), (2.5)

theempirical process of N -tuples of words. The following large deviation principle (LDP) is standard

(see e.g. Dembo and Zeitouni[14], Corollaries 6.5.15 and 6.5.17). Let

H(Q_|q⊗_ρ,N_ν):= lim

N→∞

1

Nh



Q_|

FN

(q⊗_ρ,N_ν)_|

FN

‹

be thespecific relative entropy of Q w.r.t. q_ρ⊗_,N_ν, where_FN = σ(Y(1), . . . ,Y(N)) is the sigma-algebra

generated by the first N words, Q_|

FN is the restriction of Q to FN, and h(· | ·) denotes relative

entropy (defined for probability measuresϕ,ψon a measurable spaceF ash(ϕ|ψ) =R

Flog dϕ

dψdϕ

if the density dϕ

dψ exists and as∞otherwise).

Theorem 2.1. [Annealed LDP] The family of probability distributionsP(RN ∈ ·), N ∈N, satisfies

the LDP on_Pinv(eEN)with rate N and with rate function Iann: _Pinv(EeN)_→[0,_∞]given by

Iann(Q) =H(Q_|q_ρ⊗_,N_ν). (2.7)

The rate function Iannis lower semi-continuous, has compact level sets, has a unique zero at Q=q⊗_ρ,N_ν, and is affine.

Quenched LDP.To formulate the quenched analogue of Theorem 2.1, we need some further

nota-tion. Letκ:EeN_→ENdenote theconcatenation mapthat glues a sequence of words into a sequence

of letters. ForQ∈ Pinv(EeN)such that mQ:=EQ[τ1]<∞(recall thatτ1 is the length of the first

word), defineΨ_{Q∈ P}inv_(EN

)as

Ψ_Q(_·):= 1

m_QEQ

 

τX1−1

k=0

δ_θk_κ(Y)(·)



. (2.8)

Think ofΨ_Q as the shift-invariant version of the concatenation of Y under the law Q obtained after

randomising the location of the origin.

For tr_∈N, let[_·]_tr: eE_→[Ee]_tr:=_∪tr_n=1Endenote theword length truncationmap defined by

y = (x1, . . . ,xn)7→[y]tr:= (x1, . . . ,xn∧tr), n∈N, x1, . . . ,xn∈E. (2.9)

Extend this to a map fromeENto[Ee]N tr via

(y(1),y(2), . . .)_tr:= [y(1)]_tr,[y(2)]_tr, . . ., (2.10) and to a map from_Pinv(EeN)to_Pinv([eE]N_tr)via

[Q]_tr(A):=Q(_{z_∈eEN: [z]_tr∈A_}), A_⊂[Ee]N_tr measurable. (2.11) Note that ifQ∈ Pinv_(EeN

), then[Q]_tris an element of the set

Pinv,fin(EeN) =_{Q_{∈ P}inv(eEN): m_Q<∞}. (2.12)

Theorem 2.2. [Quenched LDP, see[6], Theorem 1.2 and Corollary 1.6](a) Assume(2.1). Then, forν⊗N

–a.s. all X , the family of (regular) conditional probability distributionsP(RN ∈ · |X), N ∈N,

satisfies the LDP on _Pinv(EeN) with rate N and with deterministic rate function Ique: _Pinv(EeN) _→ [0,_∞]given by

Ique(Q):=

  

Ifin(Q), if Q_{∈ P}inv,fin(EeN), lim

tr→∞I fin _[Q]

tr

where

Ifin(Q):=H(Q|q_ρ⊗_,N_ν) + (α−1)mQH(ΨQ|ν⊗

N

). (2.14)

The rate function Ique is lower semi-continuous, has compact level sets, has a unique zero at Q=q⊗_ρ,N_ν, and is affine. Moreover, it is equal to the lower semi-continuous extension of Ifin fromPinv,fin(EeN)to

Pinv(EeN).

(b) In particular, if(2.1)holds withα=1, then forν⊗N

–a.s. all X , the familyP(RN ∈ · |X)satisfies

the LDP with rate function Ianngiven by(2.7).

Note that the quenched rate function (2.14) equals the annealed rate function (2.7) plus an

addi-tional term that quantifies the deviation ofΨ_Q from the reference lawν⊗Non the letter sequence.

This term is explicit whenmQ<∞, but requires a truncation approximation whenmQ=∞.

We close this section with the following observation. Let

Rν:=

Q_{∈ P}inv(EeN): w₋lim

L→∞

1

L

L−1

X

k=0

δ_θk_κ(Y)=ν⊗

N

Q₋a.s.

. (2.15)

be the set ofQ’s for whichthe concatenation of words has the same statistical properties as the letter sequence X. Then, forQ_{∈ P}inv,fin(eEN), we have (see[6], Equation (1.22))

Ψ_Q=ν⊗N ⇐⇒ Ique(Q) =Iann(Q) _⇐⇒ Q∈ Rν. (2.16)

3

Proof of Theorems 1.1, 1.3 and 1.6

3.1

Proof of Theorem 1.1

The idea is to put the problem into the framework of (2.1–2.5) and then apply Theorem 2.2. To that end, we pick

E:=Zd_, e_E=ÝZd _:=∪

n∈N(Zd)n, (3.1)

and choose

ν(u):=p(u), u∈Zd_{, ρ}(n):= p

2⌊n/2⌋₍₀₎

2 ¯G(0)₋1, n∈N, (3.2)

where

p(u) =p(0,u),u_∈Zd_{, p}n_{(v−u) =p}n_(u,v),u,v∈Zd_{, G}¯_{(0) =}

∞ X

n=0

p2n(0), (3.3)

the latter being the Green function ofS−S′at the origin. Recalling (1.2), and writing

zV= (z−1) +1V =1+

V

X

N=1

(z−1)N

_V

N

(3.4)

with

V N

= X

0<j1<···<jN<∞

1_{S

we have

E”_zV _|S—=1+

∞ X

N=1

(z−1)NF_N(1)(X), E”_zV— =1+

∞ X

N=1

(z−1)NF_N(2), (3.6)

with

F_N(1)(X):= X

0<j1<···<jN<∞ P S_j

1=S ′

j1, . . . ,SjN =S

′

jN |X

, F_N(2) :=EF_N(1)(X), _(3.7)

where X = (X_k)_k∈Ndenotes the sequence of increments ofS. (The upper indices 1 and 2 indicate

the number of random walks being averaged over.)

The notation in (3.1–3.2) allows us to rewrite the first formula in (3.7) as

F_N(1)(X) = X

0<j1<···<jN<∞

N

Y

i=1 pji−ji−1

 

ji−Xji−1

k=1 X_ji

−1+k

 

= X

0<j1<···<jN<∞

N

Y

i=1

ρ(j_i−j_i−1) exp

 

N

X

i=1

log

 p

ji−ji−1₍Pji−ji−1

k=1 Xji−1+k) ρ(ji−ji−1)

 

 .

(3.8)

LetY(i)= (Xji−1+1,· · ·,Xji). Recall the definition of f : ZÝ

d_{→[0,∞)in (1.5),}

f((x1, . . . ,xn)) =

pn(x₁+_{· · ·}+x_n)

p2⌊n/2⌋(0) [2 ¯G(0)−1], n∈N, x1, . . . ,xn∈Z

d_{. (3.9)}

Note that, sinceÝZd _{carries the discrete topology, f is trivially continuous.}

LetR_{N ∈ P}inv((ÝZd)N) be the empirical process of words defined in (2.5), andπ1RN ∈ P(ÝZd)the

projection ofRN onto the first coordinate. Then we have

F_N(1)(X) =E

 exp

N

X

i=1

logf(Y(i))

! X

 =E

 exp ‚ N Z Ý Zd

(π1RN)(d y)logf(y)

ΠX



, (3.10)

whereP_{is the joint law ofX andτ(recall (2.2–2.3)). By averaging (3.10) overX we obtain (recall}

the definition ofF_N(2)from (3.7))

F_N(2)=E

– exp ‚ N Z Ý Zd

(π1RN)(d y)logf(y)

Ϊ

. (3.11)

Without conditioning onX, the sequence(Y(i))_i∈Nis i.i.d. with law (recall (2.4))

q_ρ⊗_,N_ν with q_ρ,ν(x₁, . . . ,x_n) = p

2⌊n/2⌋₍₀₎

2 ¯G(0)₋1

n

Y

k=1

p(x_k), n_∈N_{, x1, . . . ,xn∈}Zd_{. (3.12)}

Next we note that

Indeed, the Fourier representation ofpn(x,y)reads

pn(x) = 1 (2π)d

Z

[−π,π)d

d k e−i(k·x)bp(k)n (3.14)

with_bp(k) =P_x∈Zdei(k·x)p(0,x). Becausep(·,·)is symmetric, we havebp(k)∈[−1, 1], and it follows

that

max

x∈Zdp

2n_{(x) =p}2n_{(0), max} x∈Zdp

2n+1_(x)

≤p2n(0), _∀n_∈N_{. (3.15)}

Consequently, f((x1, . . . ,xn)) ≤ [2 ¯G(0)−1] is bounded from above. Therefore, by applying the

annealedLDP in Theorem 2.1 to (3.11), in combination with Varadhan’s lemma (see Dembo and Zeitouni[14], Lemma 4.3.6), we getz₂=1+exp[₋r₂]with

r2:= lim

N→∞

1

NlogF

(2)

N ≤ sup Q∈Pinv₍₍ZÝd₎N

)

¨Z Ý Zd

(π1Q)(d y)logf(y)−Iann(Q)

«

= sup

q∈P(ZÝd₎

¨Z Ý Zd

q(d y)logf(y)₋h(q|q_ρ,ν)

« (3.16)

(recall (1.3–1.4) and (3.6)). The second equality in (3.16) stems from the fact that, on the set of

Q’s with a given marginalπ1Q=q, the functionQ7→Iann(Q) =H(Q|q⊗ N

ρ,ν)has a unique minimiser

Q = q⊗N

(due to convexity of relative entropy). We will see in a moment that the inequality in (3.16) actually is an equality.

In order to carry out the second supremum in (3.16), we use the following.

Lemma 3.1. Let Z:=P

y∈ZÝd f(y)qρ,ν(y). Then

Z Ý Zd

q(d y)logf(y)₋h(q|q_ρ,ν) =logZ−h(q|q∗) ∀q∈ P(ÝZd), (3.17)

where q∗(y):= f(y)q_ρ,ν(y)/Z, y_∈ÝZd_.

Proof. This follows from a straightforward computation.

Inserting (3.17) into (3.16), we see that the suprema are uniquely attained atq=q∗andQ=Q∗=

(q∗)⊗N, and thatr_2≤logZ. From (3.9) and (3.12), we have

Z=X

n∈N

X

x1,...,xn∈Zd

pn(x₁+_{· · ·}+x_n)

n

Y

k=1

p(x_k) =X

n∈N

p2n(0) =G¯(0)₋1, (3.18)

where we use that P_v∈Zdpm(u+v)p(v) = pm+1(u), u∈Zd, m ∈N, and recall that ¯G(0) is the

Green function at the origin associated withp2(_·,_·). Henceq∗is given by

q∗(x1, . . . ,xn) =

pn(x₁+_{· · ·}+x_n) ¯

G(0)₋1

n

Y

k=1

Moreover, sincez₂ =G¯(0)/[G¯(0)₋1], as noted in (1.15), we see thatz₂ =1+exp[₋logZ], i.e.,

r2=logZ, and so indeed equality holds in (3.16).

The quenched LDP in Theorem 2.2, together with Varadhan’s lemma applied to (3.8), gives z₁ = 1+exp[₋r1]with

r₁:= lim

N→∞

1

NlogF

(1)

N (X)≤ sup Q∈Pinv₍₍ZÝd₎N

)

¨Z Ý Zd

(π1Q)(d y)logf(y)−Ique(Q)

«

X₋a.s., _(3.20)

where Ique(Q) is given by (2.13–2.14). Without further assumptions, we are not able to reverse

the inequality in (3.20). This point will be addressed in Section 4 and will require assumptions (1.10–1.12).

3.2

Proof of Theorem 1.3

To compare (3.20) with (3.16), we need the following lemma, the proof of which is deferred.

Lemma 3.2. Assume (1.1). Let Q∗ = (q∗)⊗N

with q∗ as in (3.19). If mQ∗ < ∞, then Ique(Q∗) >

Iann(Q∗).

With the help of Lemma 3.2 we complete the proof of the existence of the gap as follows. Since

logf is bounded from above, the function

Q_7→

Z Ý Zd

(π1Q)(d y)logf(y)−Ique(Q) (3.21)

is upper semicontinuous. Therefore, by compactness of the level sets of Ique(Q), the function in

(3.21) achieves its maximum at someQ∗∗that satisfies

r₁=

Z Ý Zd

(π1Q∗∗)(d y)logf(y)−Ique(Q∗∗)≤

Z Ý Zd

(π1Q∗∗)(d y)logf(y)−Iann(Q∗∗)≤r2. (3.22)

Ifr1=r2, thenQ∗∗=Q∗, because the function

Q_7→

Z Ý Zd

(π1Q)(d y)logf(y)−Iann(Q) (3.23)

hasQ∗as its unique maximiser (recall the discussion immediately after Lemma 3.1). ButIque(Q∗)> Iann(Q∗)by Lemma 3.2, and so we have a contradiction in (3.22), thus arriving atr₁<r₂.

In the remainder of this section we prove Lemma 3.2.

Proof. Note that

q∗((Zd)n) = X

x1,...,xn∈Zd

pn(x₁+_{· · ·}+x_n) ¯

G(0)₋1

n

Y

k=1

p(x_k) = p

2n₍₀₎

¯

G(0)₋1, n∈N, (3.24)

and hence, by assumption (1.2),

lim

n→∞

logq∗((Zd)n)

and

m_Q∗=

∞ X

n=1

nq∗((Zd)n) =

∞ X

n=1

np2n(0) ¯

G(0)₋1. (3.26)

The latter formula shows thatmQ∗ <∞if and only ifp(·,·)is strongly transient. We will show that

m_Q∗ <∞ =⇒ Q∗= (q∗)⊗N6∈ R_ν, (3.27)

the set defined in (2.15). This implies Ψ_Q∗ 6= ν⊗N (recall (2.16)), and hence H(Ψ_Q∗|ν⊗N) > 0,

implying the claim becauseα∈(1,_∞)(recall (2.14)).

In order to verify (3.27), we compute the first two marginals ofΨ_Q∗. Using the symmetry ofp(·,·),

we have Ψ_Q∗(a) =

1

m_Q∗

∞ X

n=1

n

X

j=1

X

x1,...,xn∈Zd

x j=a

pn(x₁+_{· · ·}+x_n) ¯

G(0)₋1

n

Y

k=1

p(xk) =p(a)

P_∞

n=1np

2n−1_(a)

P_∞

n=1np2n(0)

. (3.28)

Hence,Ψ_Q∗(a) =p(a)for alla∈Zd withp(a)>0 if and only if

a_7→

∞ X

n=1

n p2n−1(a)is constant on the support ofp(_·). (3.29)

There are many p(_·,_·)’s for which (3.29) fails, and for these (3.27) holds. However, for simple

random walk (3.29) does not fail, because a _7→ p2n−1(a) is constant on the 2d neighbours of the

origin, and so we have to look at the two-dimensional marginal.

Observe thatq∗(x1, . . . ,xn) =q∗(xσ(1), . . .xσ(n))for any permutationσof{1, . . . ,n}. Fora,b∈Zd, we have

mQ∗Ψ_Q∗(a,b) =E_Q∗

 

τ1

X

k=1

1_κ(Y)k=a,κ(Y)k+1=b

 

=

∞ X

n=1 ∞ X

n′₌₁

X

x1,...,xn+n′

q∗(x₁, . . . ,x_n)q∗(x_n+1, . . . ,x_n+n′)

n

X

k=1

1_(a,b)(x_k,x_k+1)

=q∗(x1=a)q∗(x1= b) + ∞ X

n=2

(n−1)q∗ {(a,b)_{} ×}(Zd)n−2.

(3.30)

Since

q∗(x1=a) =

p(a)2 ¯

G(0)₋1+

∞ X

n=2

X

x2,...,xn∈Zd

pn(a+x₂+_{· · ·}+x_n) ¯

G(0)₋1 p(a)

n

Y

k=2 p(xk)

= p(a) ¯

G(0)₋1

∞ X

n=1

p2n−1(a)

and

q∗ _{(a,b)_{} ×}(Zd₎n−2_=1n=2 p(a)p(b)

¯

G(0)₋1p

2_(a+b)

+1n≥3

p(a)p(b) ¯

G(0)₋1

X

x3,...,xn∈Zd

pn(a+b+x3+· · ·+xn) n

Y

k=3 p(xk)

= p(a)p(b) ¯

G(0)₋1p

2n−2_(a+b),

(3.32)

we find (recall that ¯G(0)₋1=P∞_n=1p2n(0))

Ψ_Q∗(a,b) =

p(a)p(b)

h _P∞

n=1

p2n₍₀₎ih P∞ n=1

np2n₍₀₎i

‚_X∞

n=1

p2n−1(a)

_X∞

n=1

p2n−1(b)

+

_X∞

n=1 p2n(0)

_X∞

n=2

(n₋1)p2n−2(a+b)

Œ

.

(3.33)

Pick b=₋awithp(a)>0. Then, shiftingnton−1 in the last sum, we get

Ψ_Q∗(a,−a)

p(a)2 −1=

– ∞ P

n=1

p2n−1(a)

™2

h_P∞

n=1

p2n₍₀₎ih P∞ n=1

np2n₍₀₎i

>0. (3.34)

This shows that consecutive letters are not uncorrelated underΨ_Q∗, and implies that (3.27) holds as

claimed.

3.3

Proof of Theorem 1.6

The proof follows the line of argument in Section 3.2. The analogues of (3.4–3.7) are

zVe =

∞ X

N=0

(logz)N Ve

N

N!, (3.35)

with

e

VN N! =

Z ∞ 0

d t1· · ·

Z ∞

tN−1

d tN 1_{eSt1=eS′t1,...,SetN=eS′tN}, (3.36)

and

E

h

zVe |eS

i

=

∞ X

N=0

(logz)NF_N(1)(eS), E

h

zVe

i

=

∞ X

N=0

(logz)NF_N(2), (3.37)

with

F_N(1)(Se):=

Z ∞ 0

d t_{1· · ·}

Z ∞

tN−1 d t_N P

e

S_t 1=eS

′

t1, . . . ,SetN =eS

′

tN |eS

where the conditioning in the first expression in (3.37) is on the full continuous-time path Se = (Set)t≥0. Our task is to compute

er₁:= lim

N→∞

1

N logF

(1)

N (eS) Se−a.s., er2:= lim

N→∞

1

N logF

(2)

N , (3.39)

and show thater₁<er₂.

In order to do so, we write eS_t = X_J♮

t, where X

♮ _{is the discrete-time random walk with transition}

kernel p(_·,_·) and(J_t)_t≥0 is the rate-1 Poisson process on[0,_∞), and then average over the jump times of(Jt)t≥0 while keeping the jumps ofX♮fixed. In this way we reduce the problem to the one

for the discrete-time random walk treated in the proof of Theorem 1.6. For the first expression in

(3.38) thispartial annealinggives an upper bound, while for the second expression it is simply part

of the averaging overeS. Define

F_N(1)(X♮):=

Z ∞ 0

d t_{1· · ·}

Z ∞

tN−1

d t_N P(eS_t 1=Se

′

t1, . . . ,eStN =Se

′

tN |X

♮_{), F}(2)

N :=E

F_N(1)(X♮), (3.40) together with the critical values

r₁♮:= lim

N→∞

1

N logF

(1)

N (X♮) (X♮−a.s.), r

♮

2 :=_Nlim_→∞

1

N logF

(2)

N . (3.41)

Clearly,

er1≤r1♮ ander2=r2♮, (3.42)

which can be viewed as a result of “partial annealing”, and so it suffices to show thatr₁♮ <r₂♮. To this end write out

P(eSt1=Se ′

t1, . . . ,eStN =Se

′

tN |X

♮₎

= X

0≤j1≤···≤jN<∞

N

Y

i=1

e−(ti−ti−1)(ti−ti−1)

ji−ji−1

(ji−ji−1)!

!

X

0≤j₁′≤···≤j′_N<∞

N

Y

i=1

e−(ti−ti−1)(ti−ti−1)

j_i′−j′_i−1

(j_i′−j_i′₋₁)!

!  

N

Y

i=1 p_j′

i−j′i−1

 

jiX−ji−1

k=1 X♮_j

i−1+k

 

 .

(3.43)

Integrating over 0_≤t_{1≤ · · · ≤}t_N <∞, we obtain

F_N(1)(X♮) = X

0≤j1≤···≤jN<∞

X

0≤j′

1≤···≤jN′<∞

N

Y

i=1



2−(ji−ji−1)−(j′i−ji′−1)−1

[(j_i−j_i−1) + (j_i′₋j_i′₋₁)]! (ji− ji−1)!(ji′−ji′−1)!

p_j′

i−j′i−1

 

jiX−ji−1

k=1 X♮_j

i−1+k

   . (3.44) Abbreviating

Θ_n(u) =

∞ X

m=0

pm(u)2−n−m−1

_n+m

m

we may rewrite (3.44) as

F_N(1)(X♮) = X

0≤j1≤···≤jN<∞

N

Y

i=1

Θ_j

i−ji−1

 

jiX−ji−1

k=1 X♮_j

i−1+k



. (3.46)

This expression is similar in form as the first line of (3.8), except that the order of the j_i’s is not strict. However, defining

b

F_N(1)(X♮) = X

0<j1<···<jN<∞

N

Y

i=1

Θ_j

i−ji−1

 

jiX−ji−1

k=1 X♮_j

i−1+k



, (3.47)

we have

F_N(1)(X♮) =

N

X

M=0

_N

M

[Θ₀(0)]MFb_N(1₋)_M(X♮), (3.48)

with the conventionFb₀(1)(X♮)_≡1. Letting

r₁♮ = lim

N→∞

1

NlogbF

(1)

N (X

♮_{), X}♮

−a.s., (3.49)

and recalling (3.41), we therefore have the relation

r₁♮ =loghΘ₀(0) +ebr1♮

i

, (3.50)

and so it suffices to compute_br₁♮. Write

F_N(1)(X♮) =E

 exp

‚

N

Z Ý Zd

(π1RN)(d y)logf♮(y)

ΠX

♮ 

, (3.51)

where f♮: ZÝd_→[0,_∞)is defined by

f♮((x1, . . . ,xn)) =

Θ_n(x₁+_{· · ·}+x_n)

p2⌊n/2⌋(0) [2 ¯G(0)−1], n∈N, x1, . . . ,xn∈Z

d_{. (3.52)}

Equations (3.51–3.52) replace (3.8–3.9). We can now repeat the same argument as in (3.16–

3.22), with the sole difference that f in (3.9) is replaced by f♮ in (3.52), and this, combined with

Lemma 3.3 below, yields the gapr₁♮ <r₂♮.

We first check that f♮ is bounded from above, which is necessary for the application of Varadhan’s

lemma. To that end, we insert the Fourier representation (3.14) into (3.45) to obtain Θ_n(u) = 1

(2π)d

Z

[−π,π)d

d k e−i(k·u)[2_−bp(k)]−n−1, u∈Zd_{, (3.53)}

from which we see thatΘ_n(u)_≤Θ_n(0),u_∈Zd. Consequently,

f_n♮((x1,· · ·,xn))≤

Θ_n(0)

p2⌊n/2⌋(0)[2 ¯G(0)−1], n∈N, x1, . . . ,xn∈Z

Proof. For I _{⊂ {}1, . . . ,n_}, we write Ic := _{1, . . . ,n_{} \} I. For y _∈ (F _\G)Ic,z _∈ FI, we denote by (y;z) the element of F{1,...,n} defined by (y;z)_i = zi if i ∈ I, (y;z)i = yi if i ∈ Ic. Put

qI,y(z) := q(y;z)/q(ξG,I(y)), where ξG,I(y) = {(y;z′): z′ ∈ GI}, i.e., qI,y ∈ P(GI) is the law

of the coordinates in I underq given that these take values inG and that the coordinates inIc are equal to y.

FixI_{⊂ {}1, . . . ,n_}, y _∈(F_\G)Ic. We first verify that

X

z∈GI

q′(y;z)log

‚

q′(y;z)

ν⊗n_(y;z)

Œ

≤ X

z∈GI

q(y;z)log

_q(y;z)

ν⊗n_(y;z)

. (A.4)

By definition, the left-hand side of (A.4) equals

q(ξG,I(y))

X

z∈GI

Y

i∈I

νG(zi)

log q(ξG,I(y)) ν(G)|I|Q

j∈Icν(y_j)

!

=q(ξG,I(y))log

q(ξG,I(y))

ν(G)|I|Q

j∈Icν(y_j)

!

, (A.5) whereas the right-hand side of (A.4) is equal to

q(ξG,I(y))

X

z∈GI

q_I,y(z)log Q q(ξG,I(y))qI,y(z)

i∈Iν(zi)×

Q

j∈Icν(y_j)

!

. (A.6)

Thus, the right-hand side of (A.4) minus the left-hand side of (A.4) equals q(ξG,I(y))

X

z∈GI

q_I,y(z)log

‚

qI,y(z)

Q

i∈IνG(zi)

Œ

=q(ξG,I(y))h

q_I,y(_·)_|ν_G⊗|I|≥0. (A.7) The claim follows from (A.4) by observing that

h(q′|ν⊗n) = X I⊂{1,...,n}

X

y∈(F\G)I c

X

z∈GI

q′(y;z)log

‚

q′(y;z)

ν⊗n_(y;z)

Œ

, (A.8)

and analogously forh(q|ν⊗n).

For the proof of the second inequality in (4.24), i.e., H(Ψ_Qν

tr |ν

⊗N

)_≤H(Ψ_[Q]

tr |ν

⊗N

), (A.9)

we need some further notation. Let tr_∈ N be a given truncation level, _∗ a new symbol, _{∗ 6∈} E, E_∗:=E∪ {∗},Ee_∗:=_∪∞_n=0(E_∗)n, whereEe_∗0:=_{ǫ}withǫthe empty word (i.e., the neutral element of

e

E_∗viewed as a semigroup under concatenation). For y ∈eE, let

e

E_∗∋[y]_tr,∗:=

(

y, if|y|<tr,

∗tr, if_|y_{| ≥}tr, (A.10) where_∗tr=_{∗ · · · ∗}denotes the word ineE_∗consisting of tr times_∗, and

Etr∪ {ǫ} ∋[y]_tr,∼:=

(

ǫ, if|y|<tr,

LetQ_{∈ P}erg(eEN)satisfyH([Q]_tr)<∞. ForY = (Y(i))_i∈Nwith lawQandN ∈N, let

K(N,tr):=κ([Y(1)]_tr, . . . ,[Y(N)]_tr), K(N,tr,∗):=κ([Y(1)]_tr,∗, . . . ,[Y(1)]_tr,∗), K(N,tr,∼):=κ([Y(1)]_tr,∼, . . . ,[Y(1)]_tr,∼).

(A.12)

Thus, K(N,tr,∗) consists of the letters in the first N words from [Y]_tr such that letters in words of length exactly equal to tr are masked by_∗’s, whileK(N,tr,∼)consists of the letters in words of length tr among the first N words of[Y]_tr. Note that by construction there is a deterministic function

Ξ: Ee_∗×Ee→Eesuch thatK(N,tr)= Ξ(K(N,tr,∗),K(N,tr,∼)). We assume thatQ(τ1 ≥tr)>0, otherwise

K(N,tr,∼)is trivially equal toǫfor allN.

Extend [_·]_tr,∗ and [_·]_tr,∼ in the obvious way to a map on EeN and _P(EeN). Then [Q]_tr, [Q]_tr,∗,

[Q]_{tr,∼ ∈ P}erg(Ee_∗N), m[Q]tr = m[Q]tr,∗ ≤ tr, m[Q]tr,∼ = tr, Ψ[Q]tr,Ψ[Q]tr,∗,Ψ[Q]tr,∼ ∈ P erg_(EN

∗ ). By

ergodicity ofQ, we have (see[5], Section 3.1, for analogous arguments) lim

N→∞

1

NlogQ(K

(N,tr)_{) =} −m_[Q]

trH(Ψ[Q]tr) a.s. (A.13) lim

N→∞

1

NlogQ(K

(N,tr,∗)_{) =} −m_[Q]

trH(Ψ[Q]tr,∗) a.s. (A.14)

SinceQ(K(N,tr)) =Q(K(N,tr,∗),K(N,tr,∼)) =Q(K(N,tr,∗))Q(K(N,tr,∼))_|K(N,tr,∗)), we see from (A.13–A.14) that

lim

N→∞

1

NlogQ(K (N,tr,∼)₎

|K(N,tr,∗)) =₋m_[Q]

tr H(Ψ[Q]tr)−H(Ψ[Q]tr,∗)

=:₋H_tr,∼|∗(Q) a.s. (A.15) The assumption H([Q]_tr) < ∞ guarantees that all the quantities appearing in (A.13–A.15) are proper. Note that Htr,∼|∗(Q) can be interpreted as the conditional specific relative entropy of the

letters in the “long” words of[Y]_tr given the letters in the “short” words (see Lemma A.2 below). Note thatH_tr,∼|∗(Q)in (A.15) is defined as a “per word” quantity. Since the fraction of long words in

[Y]_trisQ(τ1≥tr)and each of these words contains tr letters, the corresponding conditional specific

relative entropy “per letter” isH_tr,∼|∗(Q)/[Q(τ1≥tr)tr], as it appears in (A.22) below.

Proof of (A.9). Without loss of generality we may assume thatQ(τ1≥tr)∈(0, 1). Indeed, ifQ(τ1≥

tr) =0, thenQν_tr= [Q]_tr, while ifQ(τ1≥ tr) =1, thenΨQν tr =ν

⊗N

. In both cases (A.9) obviously holds.

Step 1. We will first assume that_|E_|<∞. ThenH([Q]_tr)<∞is automatic. Sinceν⊗N

is a product measure, we have, for anyΨ_{∈ P}inv_(EN

),

H(Ψ_|ν⊗N) =₋H(Ψ)₋X x∈E

Ψ(_{x_{} ×}EN)logν(x), (A.16) whereH(Ψ)denotes the specific entropy ofΨ. We have

H(Ψ_[Q]

tr|ν

⊗N

) =₋H(Ψ_[Q]

tr)− 1 m_[Q]

tr E_Q

hτ1_X∧tr j=1

logν(Y_j(1))i,

H(Ψ_Qν tr|ν

⊗N

) =₋H(Ψ_Qν tr)−

1 m[Q]tr

E_Q

hτ1_X∧tr j=1

logν(Y_j(1));τ1<tr

i

−Q(τ1≥tr)trh(ν)

, (A.17)

whereh(ν) =₋P_x∈Eν(x)logν(x)is the entropy ofν. Hence H(Ψ_[Q]

tr|ν

⊗N

)₋H(Ψ_Qν tr|ν

⊗N

) =₋H(Ψ_[Q]

tr)−H(ΨQνtr)

− 1

m[Q]tr E_Q

h_Xtr

j=1

logν(Y_j(1));τ1≥tr

i

−Q(τ1≥tr)tr m[Q]tr

h(ν). (A.18) By (A.15) applied toQand toQν_tr (note that[Qν_tr]_tr=Qν_tr), we have

H(Ψ[Q]tr) =H(Ψ[Q]tr,∗) +

1 m[Q]tr

H_tr,∼|∗(Q), (A.19)

H(Ψ_Qν

tr) =H(Ψ[Qνtr]tr,∗) +

1 m[Qν

tr]tr

H_tr,∼|∗(Qν_tr). (A.20) By construction, m_[Qν

tr]tr = m[Q]tr, [Q ν

tr]tr,∗ = [Q]tr,∗, Htr,∼|∗(Qνtr) = Q(τ1 ≥ tr)trh(ν). Combining

(A.18–A.20), we obtain H(Ψ_[Q]

tr|ν

⊗N

)₋H(Ψ_Qν tr|ν

⊗N

) = 1

m_[Q]tr

−Htr,∼|∗(Q)−EQ

h_Xtr

j=1

logν(Y_j(1));τ1≥tr

i

. (A.21)

Finally, we observe that 1 Q(τ1≥tr)tr

−Htr,∼|∗(Q)−EQ

h_Xtr

j=1

logν(Y_j(1));τ1≥tr

i

= −Htr,∼|∗(Q)

Q(τ1≥tr)tr−

1 trEQ

h Xtr

j=1logν(Y

(1)

j )

τ1≥tr

i (A.22)

is the “specific relative entropy of the law of letters in the concatenation of long words given the concatenation of short words in[Q]_tr with respect toν⊗N

”, which is_≥0 (see Lemma A.2 below).

Step 2. We extend (A.9) to a general letter space Eby using the coarse-grainingconstruction from

[6], Section 8. LetAc ={Ac,1, . . . ,Ac,nc},c ∈N, be a sequence of nested finite partitions of E, and let_〈·〉c: E_{→ 〈}E_〉cbe the coarse-graining map as defined in [6], Section 8. Since _〈E_〉cis finite and the word length truncation[_·]_tr and the letter coarse-graining_〈·〉ccommute, we have

H(_〈Ψ_Qν tr〉c| 〈ν

⊗N

〉c)≤H(〈Ψ[Q]tr〉c| 〈ν

⊗N

〉c) for allc∈N (A.23)

by Step 1. This implies (A.9) by takingc_{→ ∞}(see the arguments in[6], Lemma 8.1 and the second part of (8.13)).

Lemma A.2. Assume _|E_| < ∞. Let tr _∈N, Q _{∈ P}erg(EeN) with Q(τ1 ≥ tr) > 0. For N ∈N, put

˜_LN:=|K(N,tr,∼)

|. Then a.s. 0_≤ lim

N→∞

1

˜_LNh Q(K (N,tr,∼)

∈ · |K(N,tr,∗))ν⊗˜LN

= −Htr,∼|∗(Q)

Q(τ1≥tr)tr−

1 trEQ

  

tr

X

j=1

logν(Y_j(1))

τ1≥tr

 

Proof. Note that, by construction, ˜L_N =˜L_N(K(N,tr,∗))is a deterministic function ofK(N,tr,∗)(namely, the number of_∗’s inK(N,tr,∗)), and

lim

N→∞

˜_{LN/N=trQ(τ1≥tr) a.s. (A.24)} by ergodicity ofQ. Fix ε >0. By ergodicity ofQ, there exists a randomN₀ <∞such that for all N _≥ N₀ there is a finite (random) set B_N,ε = B_N,ε(K(N,tr,∗)) _⊂ E˜LN _{such that Q(K}(N,tr,∼) _{∈ B}

N,ε | K(N,tr,∗))_≥1₋ε,

1

NlogQ(K

(N,tr,∼)_=b

|K(N,tr,∗))_∈−Htr,∼|∗(Q)−ε,−Htr,∼|∗(Q) +ε

(A.25) and

1 ˜ L_N

˜ LN

X

j=1

logν(b_i)_∈[χ−ε,χ+ε]withχ= 1

trEQ

Ptr

j=1logν(Y

(1)

j )

_τ1≥tr (A.26)

for allb= (b₁, . . . ,b˜_LN)∈BN,ε. Here, (A.25) follows from (A.15), while for (A.26) we note that

lim

N→∞N

−1 |K_X(N,tr)|

j=1

logν(K(_jN,tr)) =E_Q

tr_X∧τ1

j=1

logν(Y_j(1)),

lim

N→∞N

−1 |K(N,tr,∼)_|

X

j=1

logν(K(_jN,tr,∼)) =E_Q

tr

X

j=1

logν(Y_j(1));τ1≥tr

,

(A.27)

and recall (A.24). It follows that 1

˜ LN

h Q(K(N,tr,∼)_{∈ · |}K(N,tr,∗))ν⊗˜LN

= 1

˜_LN

X

b∈E˜LN

Q(K(N,tr,∼)=b|K(N,tr,∗))log

 

Q(K

(N,tr,∼)_=b

|K(N,tr,∗))

Q˜LN

j=1ν(bj)

 

=:Σ₁+ Σ₂,

(A.28)

whereΣ₁ runs over b∈BN,εandΣ2 overb∈E ˜_LN

\BN,ε. We have

Σ_1∈hχ′−ε′,χ′+ε′]withχ′=₋ Htr,∼|∗(Q)

trQ(τ1≥tr)

− 1 trEQ

Ptr

j=1logν(Y

(1)

j )

_{τ1≥tr (A.29)}

forN ≥N0 by (A.24–A.26), whereε′=ε′(Q,ε)tends to zero asε↓0. Multiplying and dividing by

Q(K(N,tr,∼)_6∈B_N,ε|K(N,tr,∗)), we see that

Σ_2≤Q(K(N,tr,∼)_6∈B_N,ε|K(N,tr,∗))max

b∈E log

₁

ν(b)

−Q(K(N,tr,∼)_6∈B_N,ε|K(N,tr,∗))logQ(K(N,tr,∼)_6∈B_N,ε|K(N,tr,∗)) +Q(K(N,tr,∼)_6∈B_N,ε|K(N,tr,∗))log_|E_|,

(A.30)

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