Variational characterization of the critical curve for pinning of random polymers

2013, Vol. 41, No. 3B, 1767–1805 DOI:10.1214/11-AOP727

©Institute of Mathematical Statistics, 2013

VARIATIONAL CHARACTERIZATION OF THE CRITICAL CURVE FOR PINNING OF RANDOM POLYMERS

B

D

C

F

H

University of Athens and Leiden University

In this paper we look at the pinning of a directed polymer by a one- dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. The- ory Related Fields 148 (2010) 403–456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases.

These variational formulas are used to obtain a necessary and sufficient cri- terion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature.

Subsequently, upper and lower bounds are derived for the inverse critical tem- perature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be nec- essary as well, a problem that we will address in a forthcoming paper.

Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valu- able insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach, that is, flexible enough to be extended to other models of random polymers with disorder.

1. Introduction and main results.

1.1. Introduction.

I. Model. Let S = (S

)

be a Markov chain on a countable state space S in which a given point is marked 0 ( N

= N ∪ {0}). Write P to denote the law of S given S

= 0 and E the corresponding expectation. Let K denote the distribution

Received January 2011; revised July 2011.

1Supported in part by the DFG-NWO Bilateral Research Group “Mathematical Models from Physics and Biology.”

MSC2010 subject classifications.Primary 60F10, 60K37; secondary 82B27, 82B44.

Key words and phrases. Random polymer, random charges, localization vs. delocalization, quenched vs. annealed large deviation principle, quenched vs. annealed critical curve, relevant vs.

irrelevant disorder, critical temperature.

1767

of the first return time of S to 0, that is,

K(n) := P(S

= 0, S

= 0 ∀0 < m < n), n ∈ N.

(1.1)

We will assume that

K(n) = 1 (i.e., 0 is a recurrent state) and

n→∞

lim

log K(n)

log n = −(1 + α) for some α ∈ [0, ∞).

(1.2)

Let ω = (ω

)

be i.i.d. R-valued random variables with marginal distribu- tion μ

. Write P = μ

to denote the law of ω, and E to denote the corresponding expectation. We will assume that

M(λ) := E(e

) < ∞ ∀λ ∈ R, (1.3)

and that μ

has mean 0 and variance 1.

Let β ∈ [0, ∞) and h ∈ R, and for fixed ω define the law P

on {0} × S

, the set of n-steps paths in S starting from 0, by putting

dP

dP

((S

)

) := 1 Z

exp

_n−1

 k=0

(βω

− h)1



1

, (1.4)

where P

is the projection of P onto {0} × S

. Here, β plays the role of the inverse temperature, h the role of the disorder bias, while Z

is the normalizing parti- tion sum. Note that k = 0 contributes to the sum, while k = n does not. Also note that the path is tied to 0 at both ends. This is done for later convenience.

R

1.1. Note that (1.2) implies p := gcd[supp(K)] = 1. If p ≥ 2, then the model can be trivially restricted to p N, so there is no loss of generality. More- over, if

K(n) < 1, then the model can be reduced to the recurrent case by a shift of h. Similarly, the restriction to μ

with mean 0 and variance 1 can be removed by a scaling of β and a shift of h.

R

1.2. The key example of the above setting is the simple random walk on Z, for which p = 2 and α =

(Spitzer [19], Section 1). In that case the process (n, S

)

can be thought of as describing a directed polymer in N

× Z, that is, pinned to the interface N

× {0} by random charges βω − h; see Figure 1.

When the polymer hits the interface at time k, it picks up a reward exp [βω

− h],

which can be either >1 or <1, depending on the value of ω

. For h ≤ 0 the polymer

tends to intersect the interface with a positive frequency (“localization”), whereas

for h > 0 large enough it tends to wander away from the interface (“delocaliza-

tion”). Simple random walk on Z

corresponds to p = 2 and α = 0, while simple

random walk on Z

, d ≥ 3, conditioned on returning to 0 corresponds to p = 2

and α =

− 1 (Spitzer [ 19], Section 1).

FIG. 1. A directed polymer sampling random charges at an interface.

II. Free energy and phase transition. The quenched free energy is defined as f

(β, h) := lim

1

n log Z

. (1.5)

Standard subadditivity arguments show that the limit exists ω-a.s. and in P-mean, and is nonrandom; see, for example, Giacomin [11], Chapter 5, and den Hol- lander [8], Chapter 11. Moreover, f

(β, h) ≥ 0 because Z

≥ e

K(n), n ∈ N, and lim

n

log K(n) = 0 by ( 1.2). The lower bound f

(β, h) = 0 is attained when S visits the state 0 only rarely. This motivates the definition of two quenched phases,

L := {(β, h) : f

(β, h) > 0 }, (1.6)

D := {(β, h) : f

(β, h) = 0},

referred to as the localized phase, respectively, the delocalized phase.

Since h → f

(β, h) is nonincreasing for every β ∈ [0, ∞), the two phases are separated by a quenched critical curve

h

(β) := inf{h : f

(β, h) = 0}, β ∈ [0, ∞), (1.7)

with L the region below the curve and D the region on and above. Since (β, h) → f

(β, h) is convex and D = {(β, h) : f

(β, h) ≤ 0} is a level set of f

, it fol- lows that D is a convex set and h

is a convex function. Since β = 0 corresponds to a homopolymer, we have h

(0) = 0; see Appendix A. It was shown in Alexan- der and Sidoravicius [2] that h

(β) > 0 for β ∈ (0, ∞). Therefore we have the qualitative picture drawn in Figure 2. We further remark that lim

h

(β)/β is finite if and only if supp(μ

) is bounded from above.

The mean value of the disorder is E(βω

−h) = −h. Thus, we see from Figure 2

that for the random pinning model localization may even occur for moderately neg-

ative mean values of the disorder, contrary to what happens for the homogeneous

pinning model, where localization occurs only for a strictly positive parameter; see

Appendix A. In other words, even a globally repulsive random interface can pin

the polymer: all that the polymer needs to do is to hit some positive values of the

disorder and avoid the negative values of the disorder.

0 β h



D L

FIG. 2. Qualitative plot of β→ h^quec (β). The fine details of this curve are not known.

The annealed free energy is defined by f

(β, h) := lim

1

n log E(Z

).

(1.8) Since

E(Z

) = E



exp

_n−1

 k=0

[log M(β) − h]1



1



(1.9) ,

we have that f

(β, h) is the free energy of the homopolymer with parameter log M(β) − h. The associated annealed critical curve

h

(β) := inf{h : f

(β, h) = 0}, β ∈ [0, ∞), (1.10)

therefore equals

h

(β) = log M(β).

(1.11)

Since f

≤ f

, we have h

≤ h

.

D

1.3. The disorder is said to be relevant for a given choice of K, μ

and β when h

(β) < h

(β), otherwise it is said to be irrelevant.

Note: In the physics literature, the term relevant disorder is reserved for the situation where the disorder not only changes the critical value but also changes the behavior of the free energy near the critical value. In the present paper we adopt the more narrow definition above.

Our main focus in the present paper will be on deriving variational formulas

for h

and h

, and on investigating under what conditions on K, μ

and β the

disorder is relevant, respectively, irrelevant.

1.2. Main results. This section contains three theorems and four corollaries, all valid subject to (1.2) and (1.3). To state these we need some further notation.

I. Notation. Abbreviate

E := supp[μ

] ⊂ R.

(1.12)

Let E

:=

E

be the set of finite words consisting of letters drawn from E.

Let P( E

) denote the set of probability measures on infinite sentences, equipped with the topology of weak convergence. Write

θ for the left-shift acting on E

, and P

( E

) for the set of probability measures that are invariant under

θ .

For Q ∈ P

( E

), let π

Q ∈ P(E) denote the projection of Q onto the first letter of the first word. Define the set

C :=

Q ∈ P

( E

) :

E

|x| d(π

Q)(x) < ∞

, (1.13)

and on this set the function

(Q) :=

E

x d(π

Q)(x), Q ∈ C.

(1.14)

We also need two rate functions on P

( E

), denoted by I

and I

, which will be defined in Section 2. These are the rate functions of the annealed and the quenched large deviation principles that play a central role in the present paper, and they satisfy I

≥ I

.

II. Theorems. With the above ingredients, we obtain the following characteri- zation of the critical curves.

T

1.4. Fix μ

and K. For all β ∈ [0, ∞), h

(β) = sup

Q∈C

[β(Q) − I

(Q) ], (1.15)

h

(β) = sup

Q∈C

[β(Q) − I

(Q) ].

(1.16)

We know that h

(β) = log M(β). However, the variational formula for h

(β) will be important for the comparison with h

(β).

Next, for β ∈ [0, ∞) define the probability measures dμ

(x) := 1

M(β) e

dμ

(x), x ∈ E, (1.17)

and

dq

(x

, x

, . . . , x

) := K(n) dμ

(x

) dμ

(x

) × · · · × dμ

(x

), (1.18)

n ∈ N, x

, x

, . . . , x

∈ E.

Further, let Q

:= q

∈ P

( E

). Then Q

is the probability measure under

which the words are i.i.d., with length drawn from K and i.i.d. letters drawn

0 β h

h

(β) h

(β)

β





FIG. 3. Uniqueness of the critical inverse temperature βc.

from μ

, while Q

differs from Q

in that the first letter of each word is drawn from the tilted probability distribution μ

. We will see that Q

is the unique max- imizer of the supremum in (1.16) [note that Q

∈ C because of ( 1.3)]. This leads to the following necessary and sufficient criterion for disorder relevance.

T

1.5. Fix μ

and K. For all β ∈ [0, ∞),

h

(β) < h

(β) ⇐⇒ I

(Q

) > I

(Q

).

(1.19)

What is appealing about (1.19) is that the gap between I

and I

needs to be established only for the measure Q

, which has a simple and explicit form. We will see that the supremum in (1.15) is attained, which is to be interpreted as saying that there is a localization strategy at the quenched critical line.

Disorder relevance is monotone in β; see Figure 3.

T

1.6. For all μ

and K there exists a β

= β

(μ

, K) ∈ [0, ∞] such that

h

(β)

= h

(β), if β ∈ [0, β

],

< h

(β), if β ∈ (β

, ∞).

(1.20)

III. Corollaries. From Theorems 1.4–1.6 we draw four corollaries. Abbreviate χ :=

n∈N

[P(S

= 0)]

, w := sup[supp(μ

) ].

(1.21)

C

1.7. If α = 0, then β

= ∞ for all μ

.

C

1.8. If α ∈ (0, ∞), then the following bounds hold:

(i) β

≥ β

with β

= β

(μ

, K) ∈ [0, ∞] given by

β

:= 0 ∨ sup{β : M(2β)/M(β)

< 1 + χ

}.

(1.22)

(ii) β

≤ β

with β

= β

(μ

, K) ∈ (0, ∞] given by β

:= inf{β : h(μ

|μ

) > h(K) }, (1.23)

where h(μ

|μ

) =

log(dμ

/dμ

) dμ

is the relative entropy of μ

w.r.t. μ

, and h(K) := −

K(n) log K(n) is the entropy of K.

C

1.9. If α ∈ (0, ∞) and χ < ∞, then β

> 0 for all μ

.

C

1.10. If α ∈ (0, ∞), then β

< ∞ for all μ

with μ

( {w}) = 0 (which includes w = ∞).

We close with a conjecture stating that the condition χ < ∞ in Corollary 1.9 is not only sufficient for β

> 0 but also necessary. This conjecture will be addressed in a forthcoming paper.

C

1.11. If α ∈ (0, ∞) and χ = ∞, then β

= 0 for all μ

. 1.3. Discussion.

I. What is known from the literature? Before discussing the results in Sec- tion 1.2, we give a summary of what is known about the issue of relevant vs.

irrelevant disorder from the literature. This summary is drawn from the papers by Alexander [1], Toninelli [20, 21], Giacomin and Toninelli [14], Derrida, Giacomin, Lacoin and Toninelli [9], Alexander and Zygouras [3, 4], Giacomin, Lacoin and Toninelli [12, 13] and Lacoin [18].

T

1.12. Suppose that condition (1.2) is strengthened to K(n) = n

L(n)

(1.24)

with α ∈ [0, ∞) and L strictly positive and slowy varying at infinity.

Then:

(1) β

= 0 when α ∈ (

, ∞).

(2) β

= 0 when α =

and lim

[log n]

L

(n) = 0 for some δ > 0.

(3) β

> 0 when α =

and

n

[L(n)]

< ∞.

(4) β

> 0 when α ∈ (0,

).

(5) β

= ∞ when α = 0.

The results in Theorem 1.12 hold irrespective of the choice of μ

; see Re- mark 1.13 below. Toninelli [21] proves that if log M(λ) ∼ Cλ

as λ → ∞ for some C ∈ (0, ∞) and γ ∈ (1, ∞), then β

< ∞ irrespective of α ∈ (0, ∞) and L.

Note that there is a small gap between cases (2) and (3) at the critical threshold

α =

.

For the cases of relevant disorder, bounds on the gap between h

(β) and h

(β) have been derived in the above cited papers subject to (1.24). As β ↓ 0, this gap decays like

h

(β) − h

(β) 

⎧⎪

⎨

⎪⎩

β

, if α ∈ (1, ∞), β

ψ (1/β), if α = 1, β

, if α ∈

, 1

(1.25)

for all choices of L, with ψ slowly varying and vanishing at infinity when L( ∞) ∈ (0, ∞).

Partial results are known for α =

. For instance, it is shown in Giacomin, Lacoin and Toninelli [13] that, under the condition in Theorem 1.12(2), the gap decays faster than any polynomial, namely, roughly like exp [−β

], β ↓ 0, when L

(n)  [log n]

, n → ∞. This implies that the disorder can at most be marginally relevant, a situation where standard perturbative arguments do not work.

R

1.13. Some of the above mentioned results are proved for Gaussian disorder only, and are claimed to be true for arbitrary disorder subject to (1.3). Full proofs for arbitrary disorder are in [9, 13, 18, 21].

R

1.14. The fact that α =

is critical for relevant vs. irrelevant dis- order is in accordance with the so-called Harris criterion for disordered systems (see Harris [17]): “Arbitrary weak disorder modifies the nature of a phase transi- tion when the order of the phase transition in the nondisordered system is < 2.”

The order of the phase transition for the homopolymer, which is briefly described in Appendix A, is < 2 precisely when α ∈ (

, ∞) (see Giacomin [11], Chapter 2).

This link is emphasized in Toninelli [20].

II. What is new in the present paper? The main importance of our results in Section 1.2 is that they open up a new window on the random pinning problem.

Whereas the results cited in Theorem 1.12 are derived with the help of a variety of estimation techniques, like fractional moment estimates and trial choices of local- ization strategies, Theorem 1.4 gives a variational characterization of the critical curves, that is, new. (It is very rare indeed that critical curves for disordered sys- tems allow for a direct variational representation.) Theorem 1.5 gives a necessary and sufficient criterion for disorder relevance that, although not easy to handle, at least is explicit and offers a different handle. Theorem 1.6 shows that unique- ness of the inverse critical temperature is a direct consequence of this criterion, while Corollaries 1.7–1.10 show that the criterion can be used to obtain important information on the inverse critical temperature.

R

1.15. Theorem 1.6 was proved in Giacomin, Lacoin and

Toninelli [13] with the help of the FKG-inequality.

R

1.16. Corollary 1.7 is the main result in Alexander and Zy- gouras [4].

R

1.17. Since (see Section 8) lim

M(2β)/M(β)

= 1, lim

β→∞

h(μ

|μ

) = log[1/μ

( {w})], (1.26)

with the understanding that the second limit is ∞ when μ

( {w}) = 0, Corollary 1.8 implies Corollaries 1.9 and 1.10. Corollary 1.10 was noted also in Alexander and Zygouras [4].

R

1.18. Note that χ = E(|I

∩I

|) with I

, I

two independent copies of the set of return times of S [recall (1.1)]. Thus, according to Corollary 1.9 and Conjecture 1.11, β

> 0 is expected to be equivalent to the renewal process of joint return times to be recurrent. Note that 1/P(I

∩ I

= ∅) = 1 + χ

(see Spitzer [19], Section 1), the quantity appearing in Corollary 1.8(i).

R

1.19. If μ

is Bernoulli(1/2) on {−1, 1}, (1.26) gives that lim

h(μ

|μ

) = log 2. For any α > 0, we can find a distribution K that sat- isfies (1.2) and H (K) < log 2, and thus (1.23) implies that β

= β

(μ

, K) < ∞.

This shows that for α > 0, the condition μ

( {w}) = 0 is not (!) necessary for β

< ∞.

R

1.20. As shown in Doney [10], subject to the condition of regular variation in (1.24),

P(S

= 0) ∼ C

n

L(n) (1.27)

as n → ∞ with C

= (α/π) sin(απ) when α ∈ (0, 1).

Hence the condition χ < ∞ in Corollary 1.9 is satisfied exactly for α ∈ (0,

) and L arbitrary, and for α =

and

n

[L(n)]

< ∞. This fits precisely with cases (3) and (4) in Theorem 1.12.

R

1.21. Corollary 1.8(ii) is essentially Corollary 3.2 in Toninelli [21], where the condition for relevance, h(μ

|μ

) > h(K), is given in an equivalent form (see equation (3.6) in [21]). Note that, by (1.2), h(K) < ∞ when α ∈ (0, ∞).

1.4. Outline. In Section 2 we formulate the annealed and the quenched large

deviation principles (LDP) that are in Birkner, Greven and den Hollander [6],

which are the key tools in the present paper. In Section 3 we use these LDP’s to

prove Theorem 1.4. In Section 4 we compare the variational formulas for the two

critical curves and prove the criterion for disorder relevance stated in Theorem 1.5.

FIG. 4. Cutting words out from a sequence of letters according to renewal times.

In Section 5 we reformulate this criterion to put it into a form, that is, more con- venient for computations. In Section 6 we use the latter to prove Theorem 1.6. In Sections 7–8 we prove Corollaries 1.7–1.10. Appendix A collects a few standard facts about the homopolymer, while Appendix B provides the details of the proof of a key lemma in Section 3 based on an approximation argument in [6].

2. Annealed and quenched LDP. In this section we recall the main results from Birkner, Greven and den Hollander [6] that are needed in the present paper.

Section 2.1 introduces the relevant notation, while Sections 2.2 and 2.3 state the relevant annealed and quenched LDP’s.

2.1. Notation. Let E be a Polish space, playing the role of an alphabet, that is, a set of letters. Let E

:=

E

be the set of finite words drawn from E, which can be metrized to become a Polish space.

Fix μ

∈ P(E), and K ∈ P(N) satisfying ( 1.2). Let X = (X

)

be i.i.d. E- valued random variables with marginal law μ

, and τ = (τ

)

i.i.d. N-valued random variables with marginal law K. Assume that X and τ are independent, and write P

to denote their joint law. Cut words out of the letter sequence X according to τ (see Figure 4), that is, put

T

:= 0 and T

:= T

+ τ

, i ∈ N, (2.1)

and let

Y

:= (X

, X

, . . . , X

), i ∈ N.

(2.2)

Under the law P

, Y = (Y

)

is an i.i.d. sequence of words with marginal dis- tribution q

on E

given by

dq

(x

, . . . , x

)

:= P

Y

∈ (dx

, . . . , dx

)

(2.3)

= K(n) dμ

(x

) × · · · × dμ

(x

), n ∈ N, x

, . . . , x

∈ E.

The reverse operation of cutting words out of a sequence of letters is glueing

words together into a sequence of letters. Formally, this is done by defining a con-

catenation map κ from E

to E

. This map induces in a natural way a map from

P( E

) to P(E

), the sets of probability measures on E

and E

(endowed with

the topology of weak convergence). The concatenation q

◦ κ

of q

equals

μ

, as is evident from (2.3)

2.2. Annealed LDP. Let P

( E

) be the set of probability measures on E

that are invariant under the left-shift

θ acting on E

. For N ∈ N, let (Y

, . . . , Y

)

be the periodic extension of the N -tuple (Y

, . . . , Y

) ∈ E

to an element of E

, and define

R

:= 1 N

N−1 i=0

δ

∈ P

( E

).

(2.4)

This is the empirical process of N -tuples of words. The following annealed LDP is standard; see, for example, Dembo and Zeitouni [7], Section 6.5. For Q ∈ P

( E

), let H (Q |q

) be the specific relative entropy of Q w.r.t. q

de- fined by

H (Q|q

) := lim

N→∞

1

N h(π

Q|π

q

), (2.5)

where π

Q ∈ P( E

) denotes the projection of Q onto the first N words, h( ·|·) denotes relative entropy, and the limit is nondecreasing.

T

2.1. The family P

(R

∈ ·), N ∈ N, satisfies the LDP on P

( E

) with rate N and with rate function I

given by

I

(Q) := H(Q|q

), Q ∈ P

( E

).

(2.6)

This rate function is lower semi-continuous, has compact level sets, has a unique zero at q

, and is affine.

2.3. Quenched LDP. To formulate the quenched analog of Theorem 2.1, we need some more notation. Let P

(E

) be the set of probability measures on E

that are invariant under the left-shift θ acting on E

. For Q ∈ P

( E

) such that m

:= E

(τ

) < ∞ (where E

denotes expectation under the law Q and τ

is the length of the first word), define



:= 1 m

E

_τ1−1

 k=0

δ



∈ P

(E

).

(2.7)

Think of 

as the shift-invariant version of Q ◦ κ

obtained after randomiz- ing the location of the origin. This randomization is necessary because a shift- invariant Q in general does not give rise to a shift-invariant Q ◦ κ

.

For tr ∈ N, let [·]

: E

→ [ E]

=

E

denote the truncation map on words defined by

y = (x

, . . . , x

) → [y]

:= (x

, . . . , x

), n ∈ N, x

, . . . , x

∈ E, (2.8)

that is, [y]

is the word of length ≤ tr obtained from the word y by dropping all

the letters with label > tr. This map induces in a natural way a map from E

to

[ E

]

, and from P

( E

) to P

( [ E

]

). Note that if Q ∈ P

( E

), then [Q]

is an element of the set

P

( E

) = {Q ∈ P

( E

) : m

< ∞}.

(2.9)

T

2.2. (Birkner, Greven and den Hollander [6]) Assume (1.2). Then, for μ

-a.s. all X, the family of (regular) conditional probability distributions P

(R

∈ ·|X), N ∈ N, satisfies the LDP on P

( E

) with rate N and with deter- ministic rate function I

given by

I

(Q) :=



I

(Q), if Q ∈ P

( E

),

tr

lim

I

( [Q]

), otherwise, (2.10)

where

I

(Q) := H(Q|q

) + αm

H (

|μ

).

(2.11)

This rate function is lower semi-continuous, has compact level sets, has a unique zero at q

and is affine.

There is no closed form expression for I

(Q) when m

= ∞. For later refer- ence we remark that, for all Q ∈ P

( E

),

I

(Q) = lim

tr→∞

I

( [Q]

) = sup

tr∈N

I

( [Q]

), (2.12)

I

(Q) = lim

tr→∞

I

( [Q]

) = sup

tr∈N

I

( [Q]

)

as shown in [6], Lemma A.1. A remarkable aspect of (2.11) in relation to (2.6) is that it quantifies the difference between I

and I

. Note the explicit appearance of the tail exponent α. Also note that I

= I

when α = 0.

3. Variational formulas: Proof of Theorem 1.4. In Section 3.1 we prove (1.16), the variational formula for the annealed critical curve. The proof of (1.15) in Sections 3.2–3.4, the variational formula for the quenched critical curve, is longer. In Section 3.2 we first give the proof for μ

with finite support.

In Section 3.3 we extend the proof to μ

satisfying (1.3). In Section 3.4 we prove three technical lemmas that are needed in Section 3.3.

3.1. Proof of (1.16).

P

. Recall from (1.17) and (1.18) that Q

= q

, and from (1.11) that h

(β) = log M(β). Below we show that for every Q ∈ P

( E

),

β(Q) − I

(Q) = log M(β) − H(Q|Q

).

(3.1)

Taking the supremum over Q, we arrive at (1.16). Note that the unique probability measure that achieves the supremum in (3.1) is Q

, which is an element of the set C defined in (1.13) because of (1.3).

To get (3.1), note that H (Q |Q

) is the limit as N → ∞ of [recall ( 1.17) and (1.18)]

1 N

 E^N

log



d(π

Q)

d(π

Q

) (y

, . . . , y

)



d(π

Q)(y

, . . . , y

)

= 1 N

 E^N

log



d(π

Q)

d(π

Q

) (y

, . . . , y

)

× M(β)

e



d(π

Q)(y

, . . . , y

) (3.2)

= log M(β) + 1

N h(π

Q |π

Q

)

− β 1 N

E^N

[c(y

) + · · · + c(y

) ] d(π

Q)(y

, . . . , y

),

where, c(y) denotes the first letter of the word y. In the last line of (3.2), the limit as N → ∞ of the second quantity is H(Q|Q

) = I

(Q), while the integral equals N (Q) by shift-invariance of Q. Thus, (3.1) follows. 

3.2. Proof of (1.15) for μ

with finite support.

P

. The proof comes in three steps.

Step 1: An alternative way to compute the quenched free energy f

(β, h) from (1.5) is through the radius of convergence z

(β, h) of the power series

 n∈N

z

Z

, (3.3)

because

z

(β, h) = e

. (3.4)

Write

Z

=

N∈N

 0=k0<k₁<···<kN=n

N i=1

K(k

− k

)e

, (3.5)

so that, for z ∈ (0, ∞),

 n∈N

z

Z

=

N∈N

F

(z), (3.6)

where we abbreviate

F

(z) :=

0=k0<···<kN<∞

N i=1

z

K(k

− k

)e

.4

(3.7)

Step 2: We return to the setting of Section 2. The letter space is E, the word space is E

=

E

, the sequence of letters is ω = (ω

)

, while the sequence of renewal times is (T

)

= (k

)

. Each interval I

:= [k

, k

) of integers cuts out a word ω

:= (ω

, . . . , ω

). Let

R

= R

((k

)

) := 1 N

N−1 i=0

δ

(3.8)

denote the empirical process of N -tuples of words in ω cut out by the first N renewals. Then we can rewrite F

(z) as

F

(z) = E



exp



N

 E



τ (y) log z +

βc(y) − h

d(π

R

)(y)



(3.9)

= e

E

exp [Nm

log z + Nβ(R

)]

,

where τ (y) and c(y) are the length, respectively, the first letter of the word y, π

R

is the projection of R

onto the first word, while m

N

and (R

) are the average word length, respectively, the average first letter of the first word un- der R

.

To identify the radius of convergence of the series in the left-hand side of (3.6), we apply the root test for the series in the right-hand side of (3.6) using the expres- sion in (3.9). To that end, let

S

(β ; z) := lim sup

N→∞

1

N log E

exp[Nm

log z + Nβ(R

)]

. (3.10)

Then

lim sup

N→∞

1

N log F

(z) = −h + S

(β; z).

(3.11)

We know from (3.4) and the nonnegativity of f

(β, h) that z

(β, h) ≤ 1, and we are interested in knowing when it is < 1, respectively, = 1 [recall (1.6)]. Hence, the sign of the right-hand side of (3.11) for z ↑ 1 will be important as the next lemma shows.

L

3.1. For all β ∈ [0, ∞) and h ∈ R,

S

(β ; 1−) < h ⇒ f (β, h) = 0, (3.12)

S

(β ; 1−) > h ⇒ f (β, h) > 0.

P

. The first line holds because, by (3.11), −h+S

(β ; 1−) < 0 implies

that the sums in (3.6) converge for |z| < 1, so that z

(β, h) ≥ 1, which gives

f

(β, h) ≤ 0. The second line holds because if −h+S

(β ; 1−) > 0, then there

exists a z

< 1 such that −h+S

(β; z

) > 0, which implies that the sums in (3.6)

diverge for z = z

, so that z

(β, h) ≤ z

< 1, which gives f

(β, h) > 0. 

0 z S

(β ; z)

h

(β)

1

∞





FIG. 5. Qualitative plot of z→ S^que(β; z).

Lemma 3.1 implies that

h

(β) = S

(β ; 1−).

(3.13)

The rest of the proof is devoted to computing S

(β ; 1−).

Step 3: Since μ

has finite support, Q → (Q) is continuous. Therefore we can apply Varadhan’s lemma to the expression in (3.10) for z = 1 using the LDP of Theorem 2.2. This gives

S

(β ; 1) = sup

Q∈P^inv(E^N)

[β(Q) − I

(Q) ].

(3.14)

We would like to do the same for (3.10) with z < 1, and subsequently take the limit z ↑ 1, to get (see Figure 5)

S

(β ; 1−) = sup

Q∈P^inv(E^N)

[β(Q) − I

(Q) ].

(3.15)

However, even though Q → (Q) is continuous (because μ

has finite support), Q → m

is only lower semicontinuous. Therefore we proceed by first showing that the term N m

N

log z in (3.10) is harmless in the limit as z ↑ 1.

L

3.2. S

(β ; 1−) = S

(β ; 1) for all β ∈ [0, ∞).

P

. Since S

(β ; 1−) ≤ S

(β ; 1), we need only prove the reverse in- equality. The idea is to show that, for any Q ∈ P

( E

) and in the limit as N → ∞, R

can be arbitrarily close to Q with probability ≈ exp[−NI

(Q) ] while m

N

remains bounded by a large constant. Therefore, letting N → ∞ fol- lowed by z ↑ 1, we can remove the term Nm

log z in (3.10). The details are given in Appendix B. 

Combining Lemma 3.2 with (3.13) and (3.14), we obtain (1.15). 

3.3. Proof of (1.15) for μ

satisfying (1.3). The proof stays the same up to (3.13). Henceforth write C = C(μ

) to exhibit the fact that the set C in ( 1.13) depends on μ

via its support E in (1.12), and define

A(β) := sup

Q∈C(μ₀)

[β(Q) − I

(Q)], (3.16)

which replaces the right-hand side of (3.15). We will show the following.

L

3.3. S

(β ; 1−) = A(β) for all β ∈ (0, ∞).

P

. The proof of the lemma is accomplished in four steps. Along the way we use three technical lemmas, the proof of which is deferred to Section 3.4. Our starting point is the validity of the claim for μ

with finite support obtained in Lemma 3.2. (Note that |E| < ∞ implies C = C(μ

) = P

( E

).)

Step 1: S

(β ; 1−) ≤ A(β) for all β ∈ (0, ∞) when μ

satisfies (1.3).

P

. We have S

(β ; 1−) ≤ S

(β ; 1). We will show that S

(β ; 1) ≤ A(pβ)/p for all p > 1. Taking p ↓ 1 and using the continuity of A, proven in Lemma 3.4 below, we get the claim.

For M > 0, let



(Q) :=

E

(x ∧ M) d(π

Q)(x).

(3.17)

Then, for any p, q > 1 such that p

+ q

= 1, we have E

e

= E

e

e

≤

E

e

E

e

(3.18)

≤

E

e

E

e

,

where y

, . . . , y

are the N words determining R

and c(y

) is the first letter of the ith word. Hence

1

N log E

e

≤ 1 p

1

N log E

e

(3.19)

+ 1 q

1

N log E

e

. Since Q → 

(Q) is upper semicontinuous, Varadhan’s lemma gives

lim sup

N→∞

1

N log E

e

≤ sup

Q∈P^inv(E^N)

[pβ

(Q) − I

(Q) ].

(3.20)

Clearly, Q’s with

(x ∧ 0) d(π

Q)(x) = −∞ do not contribute to the supre-

mum. Also, Q’s with

(x ∨ 0) d(π

Q)(x) = ∞ do not contribute, because for

such Q we have I

(Q) = ∞, by Lemma 3.5 below, and 

(Q) < ∞. Since



≤ , we therefore have sup

Q∈P^inv(E^N)

[pβ

(Q) − I

(Q)] ≤ sup

Q∈C(μ₀)

[pβ(Q) − I

(Q)]

(3.21)

= A(pβ).

Next, we use the following observation. For any sequence  = (

)

of positive random variables on a space with probability measure P, we have

lim sup

N→∞

1

N log 

≤ lim sup

N→∞

1

N log E(

) P-a.s., (3.22)

by the first Borel–Cantelli lemma. Applying this to



:= E

e

_N

i=1c(y_i)1{c(yi)>M}

(3.23)

with E(

) =

E

e

dμ

(x)

N

=: (c

)

, we get, after letting N → ∞ in ( 3.19),

S

(β ; 1) ≤ 1

p A(pβ) + 1

q log c

. (3.24)

By (1.3), we have c

< ∞ for all M > 0 and lim

c

= 1. Hence S

(β ; 1) ≤ A(pβ)/p. 

Step 2: S

(β ; 1−) ≥ A(β) for all β ∈ (0, ∞) when μ

has bounded support.

P

. In the estimates below, we abbreviate L

:= Nm

, (3.25)

the sum of the lengths of the first N words. The proof is based on a discretization argument similar to the one used in [6], Section 8. For δ > 0 and x ∈ E, let x

:=

sup {kδ : k ∈ Z, kδ ≤ x}. The operation · extends to measures on E, E

and E

in the obvious way. Now, R



satisfies the quenched LDP with rate function I

, the quenched rate function corresponding to the measure μ



. Clearly,

E

e

≥ E

e

, (3.26)

and so, by the results in Section 3.2, we have S

(β ; 1−) ≥ sup

Q∈C(μ0δ)

[β(Q) − I

(Q) ].

(3.27)

For every Q ∈ C(μ

), we have

(Q) = lim

δ↓0

( Q

), I

(Q) = lim

I

n

( Q

),

(3.28)

where δ

= 2

. The first relation holds because ( Q

) ≤ (Q) ≤ (Q

) +δ, the second relation uses Lemma 3.6(i) below. Hence the claim follows by picking δ = δ

in (3.27) and letting n → ∞. 

Step 3: S

(β ; 1−) ≥ A(β) for all β ∈ (0, ∞) when μ

satisfies (1.3) with support bounded from below.

P

. For M > 0 and x ∈ E, let x

= x ∧ M. This truncation operation acts on μ

by moving the mass in (M, ∞) to M, resulting in a measure μ

with bounded support and with associated quenched rate function I

. Let R

be the empirical process of N -tuples of words obtained from R

defined in (2.4) after replacing each letter x ∈ E by x

. We have

E

e

≥ E

e

. (3.29)

Combined with the result in Step 2, this bound implies that S(β ; 1−) ≥ sup

Q∈C(μ^M₀ )

[β(Q

) − I

(Q

) ].

(3.30)

For every Q ∈ C(μ

), we have

(Q) = lim

M→∞

(Q

) = lim

M→∞

E

(x ∧ M) d(π

Q)(x), (3.31)

I

(Q) = lim

M→∞

I

(Q

).

The first relation holds by dominated convergence, and the second relation uses Lemma 3.6(ii) below. It follows from (3.31) that

lim sup

M→∞

sup

Q∈C(μ^M₀ )

[β(Q

) − I

(Q

) ] ≥ β(Q) − I

(Q) (3.32)

∀Q ∈ C(μ

), which combined with (3.30) yields

S(β ; 1−) ≥ β(Q) − I

(Q) ∀Q ∈ C(μ

).

(3.33)

Take the supremum over Q ∈ C(μ

) to get the claim. 

Step 4: S

(β ; 1−) ≥ A(β) for all β ∈ (0, ∞) when μ

satisfies (1.3).

P

. For M > 0 and x ∈ E, let x

= x ∨ (−M). This truncation opera-

tion acts on μ

by moving the mass in ( −∞, −M) to −M, resulting in a measure

μ

with support bounded from below and with associated quenched rate func-

tion I

. Let R

be the empirical process of N -tuples of words obtained

from R

defined in (2.4) after replacing each letter x ∈ E by x

.