Localization of directed polymers with general reference walk

Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an $\alpha$-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer's endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.


Introduction
The probabilistic model of directed polymers in random environment was introduced by Imbrie and Spencer [34] as a reformulation of Huse and Henley's approach [33] to studying the phase boundary of the Ising model in the presence of random impurities. In its classical form, the model considers a simple random walk (SRW) on the integer lattice Z d , whose paths-considered the "polymer"-are reweighted according to a random environment that refreshes at each time step. Large values in the environment tend to attract the random walker and possibly force localization phenomena; this attraction grows more effective in lower dimensions and at lower temperatures. On the other hand, the random walk's natural dynamics favor diffusivity. Which of these competing features dominates asymptotically is a central question in the study of directed polymers. Much progress has been made over the last thirty years in understanding polymer behavior; for a comprehensive and up-to-date survey, the reader is referred to the recent book by Comets [23].
In [22], Comets initiated the study of long-range directed polymers. In this model, the simple random walk is replaced by a general random walk capable of superdiffusive motion. More specifically, it is assumed that the walk belongs to the domain of attraction of an α-stable law for some α ∈ (0, 2]. For example, any walk having increments with a finite second moment belongs to the α = 2 case. Under this assumption the longrange polymer can model the behavior of heavy-tailed walks, such as Lévy flights, when placed in an inhomogeneous random environment. Indeed, Lévy flights in random potentials have been used to study chemical reactions [19] and particle dispersions [57,14]. Moreover, their continuous-time analogs, Lévy processes, appear in a variety of disciplines including fluid mechanics, solid state physics, polymer chemistry, and mathematical finance [8]. This is relevant because α-stable polymers are known to obey a scaling CLT at sufficiently high temperatures ( [22,Theorem 4.2] and [66,Theorem 1.9]), which generalizes the Brownian CLT proved in [29,Theorem 1.2] when the reference walk is SRW.
Interestingly, universal behaviors have also appeared at low temperatures, where the system exists in a "disordered" phase. Part of the work in [22,66] was to extend localization results known for polymers constructed from a SRW to those constructed with α-stable reference walks. This paper continues the advance in this direction by proving that in the localization regime, certain qualitative behaviors of the polymer's endpoint distribution are the same for any reference walk. Namely, the strongest forms of localization known for arbitrary environment and arbitrary dimension, which were only recently proved for the SRW case in [10], are established here for the general case.
The organization of the remaining introduction is as follows. After the polymer model is formally introduced, we will recall the relevant facts from the literature in Section 1.2 and state our main results in Section 1.3. The proof strategy is outlined in Section 1.4, which describes how the approach used in [10] must be expanded to work for general polymers. Finally, Section 1.5 offers references on other fronts of progress in both the short-and the long-range settings.

The model
Let d be a positive integer, to be called the spatial or transverse dimension. Let P denote the law of the reference walk, which is a homogeneous random walk (ω i ) i≥0 on Z d . To be precise, we state the assumptions on P : P (ω 0 = 0) = 1, P (ω i+1 = x | ω i = y) = P (ω 1 = x − y) =: P (y, x) < 1. (1.1) Next we introduce a collection of i.i.d. random variables η = (η(i, x) : i ≥ 1, x ∈ Z d ), called the random environment, supported on some probability space (Ω, F, P). We will write E and E for expectation with respect to P and P, respectively. Finally, let β > 0 be a parameter representing inverse temperature. Then for n ≥ 0, the quenched polymer measure of length n is the Gibbs measure ρ n defined by Polymers with general reference walk The normalizing constant Z n := E(e −βHn(ω) ) = x1,..., xn∈Z d P (x i−1 , x i ), x 0 := 0, is called the quenched partition function. A fundamental quantity of the system is calculated from this constant, namely the quenched free energy, F n := log Z n n .
We specify "quenched" to indicate that the randomness from the environment has not been averaged out. That is, ρ n , Z n , and F n are each random processes with respect to the filtration F n := σ(η(i, x) : 1 ≤ i ≤ n, x ∈ Z d ), n ≥ 0.
When Z n is replaced by its expectation E(Z n ), one obtains the annealed free energy, log E(Z n ) n = log(E e βη ) n n = log E(e βη ) =: λ(β), where η denotes (here and henceforth) a generic copy of η(1, 0). Notice that λ(·) depends only on the law of the environment, which we denote L η . We will assume finite (1 + ε)exponential moments, 0 < β < β max := sup{t ≥ 0 : λ(±t) < ∞}, (1.2) so that Z n has finite ±(1 + ε)-moments at the given inverse temperature. Otherwise L η is completely general, although to avoid trivialities, we will always assume that η is not an almost sure constant, and that β is strictly positive.

Overview of known results
Two fundamental facts are the convergence of the free energy and the existence of a corresponding phase transition. The first result below was initially shown by Carmona  Theorem A ([26, Proposition 2.5]). Let P be SRW and assume λ(t) < ∞ for all t ∈ R. Then there exists a deterministic constant p(β) such that lim n→∞ F n = p(β) a.s. Theorem B ([29, Theorem 3.2(b)]). Let P be SRW and assume λ(t) < ∞ for all t ∈ R. Then there exists a critical value β c ∈ [0, ∞] such that 0 ≤ β ≤ β c ⇒ p(β) = λ(β) β > β c ⇒ p(β) < λ(β).

Polymers with general reference walk
Theorem C ( [26,Corollary 2.2]). Let P be SRW and assume λ(t) < ∞ for all t ∈ R.
Define the (random) set Then p(β) < λ(β) if and only if there exists ε > 0 such that In [62], A ε i is called the set of ε-atoms. In other words, a polymer is localized when it contains macroscopic atoms that persist with positive density. This result was generalized to the case of 0 < β < B := sup{t ≥ 0 : λ(t) < ∞} by Vargas [62,Theorem 3.6], whose argument was extended to general P by Wei [ ρ i (ω i ∈ A ε i ) ≥ δ a.s. (1.4) We conclude this section by recalling a simple sufficient condition for localization. Notice that the condition depends only on the entropy of the random walk increment, not its actual distribution. One can therefore interpret the result as providing a temperature threshold below which concentration of the polymer measure is strong enough to overcome any superdiffusivity of the random walk.

Results of this paper
The temperature range at which localization occurs depends on P , but to the extent described in our two main results below, the type of localization does not. We will also prove Theorems A and B in the general setting so that the statements of localization results make sense. First, it was recently shown in [10] that β 0 in (1.4) can be taken equal to β c . Alternatively, the right-hand side of (1.3) can be taken arbitrarily close to 1. These realizations continue to hold true when P is any walk satisfying (1.1), as demonstrated in the following improvement of Theorems C and D, thus extending [10, Theorem 1.1] beyond the SRW case. (a) If p(β) < λ(β), then for every sequence ε i → 0, Polymers with general reference walk This result is later stated and proved as Theorem 5.3. In case (a) above, we say that the sequence (ρ i (ω i ∈ ·)) i≥0 is asymptotically purely atomic 1 , which is meant to indicate that all of the limiting mass is atomic, not just an ε-fraction.
(a) If p(β) < λ(β), then for any δ > 0, there exists K < ∞ and θ > 0 such that , then for any δ ∈ (0, 1) and any K > 0, This result appears as Theorem 5.4 in the sequel. In case (a), we say that the sequence of measures (ρ i (ω i ∈ ·)) i≥0 exhibits geometric localization with positive density. The constant θ gives a lower bound on the density of endpoint distributions displaying the desired level of concentration. If θ can be taken equal to 1, then essentially all large atoms remain close, so we would say the sequence is geometrically localized with full density. That this is the case is sometimes called the "favorite region conjecture", although one possibility is that it is true only in low dimensions. The only model for which it is known to be true is the one-dimensional log-gamma polymer [56], whose exact solvability was leveraged in [25] to obtain a limiting law for the endpoint distribution. The general method of [10] also was to establish a limiting law for the endpoint distribution, but in a more abstract compactification space. A sufficient condition for the existence of a favorite region was given [10,Theorem 7.3(b)], although a way to check the condition is not currently known.

Comment on moment assumptions
While Theorems 1.1 and 1.2 were shown in [10] in the case P is SRW, that paper also critically assumed λ(±2β) < ∞. Here we are establishing the results not only for an arbitrary reference walk, but also under the weaker hypothesis (1.2). In this way, the present study both expands previous results to the setting of a general walk, and optimizes assumptions even in the classical case. While this latter point may seem minor, it actually permits parameter ranges for which the regions of interest had been 1 To the author's knowledge, this terminology was first used by Vargas The assumption λ(±2β) < ∞ only covers β < 1 2 . By Theorem E, we know that there is localization (in the SRW case) at any β such that βλ (β) − λ(β) > log(2d). But for any positive integer d (in particular d ≥ 3, where localization does not necessarily occur), Therefore, the temperature regime for which localization is actually known has no intersection with the regime in which hypothesis λ(±2β) < ∞ is true. This paper eliminates this discrepancy by assuming only (1.2). The technical challenges incurred are non-trivial, but the fact that they can be overcome reflects the generality with which our methodology may be useful, possibly in contexts other than polymer models. Indeed, this feature is itself a motivation for the present study.

Outline of methods
We write f n to denote the probability mass function on Z d of the (random) endpoint distribution ρ n (ω = ·) for the length-n polymer. It is not difficult to see that (f n ) n≥0 is an 1 (Z d )-valued Markov process with respect to the canonical filtration (F n ) n≥0 (c.f. Section 3.3). Unfortunately, the space 1 (Z d ) is too large to establish any type of convergence. More to the point, we cannot expect any tightness for (ρ n ) n≥0 .
In [10], this issue is resolved by constructing a compact space in which to embed the Markov chain. Specifically, the mass functions on Z d are identified with mass functions on N × Z d , which themselves are part of a larger space: A pseudometric d * can be defined on S 0 such that the resulting quotient space S := S 0 /(d * = 0) is compact. After not too much work, one can show that the equivalence classes under d * are the orbits under translations. In other words, the only information that is lost by passing from 1 (Z d ) to S is the location of the origin. We call the elements of S partitioned subprobability measures.
In hindsight, then, the only obstructions to compactness were translations, which form a non-compact group under which Z d is invariant. Observed also for more general unbounded domains, this phenomenon is often called "concentration compactness" in the study of PDEs and calculus of variations. In the 1980s, Lions [39][40][41][42] made highly effective use of the concept by transferring problems to compact spaces using concentration functions introduced by Lévy [38], although the idea for this type of compactification scheme goes back to work of Parthasarathy, Ranga Rao and Varadhan [53]. The particular construction in [10] was inspired in part by a continuum version used by Mukherjee and Varadhan [46] to prove large deviation principles for Brownian occupation measures.
Once the Markov chain of endpoint distributions is embedded in a compact space, a few key ideas can be wielded to great effect: (i) Consider the "update map" T which receives a starting position for the chain and outputs the law after one step. Provided T is continuous, the chain's empirical measure will converge to stationarity.
(ii) A convexity argument shows that the starting configuration of P (ω 0 = 0) = 1 is optimal in yielding the minimal expected free energy. (iii) The two previous points imply that the chain's empirical measure must converge to energy-minimizing stationarity. This fact provides a variational formula (4.8) for the limiting free energy. Furthermore, by examining what properties an energyminimizing stationary distribution must have, we can deduce certain asymptotic properties of the chain. In particular, Theorems 1.1 and 1.2 will follow, as described in Sections 4 and 5.
Two of the most challenging steps are (a) constructing the compact space; and (b) proving continuity of T . Although the remaining work to prove Theorems 1.1 and 1.2 can largely be taken from [10], these two parts must undergo non-trivial generalizations. This is done in Section 3.
First, the construction in [10] of the compact metric space (S, d * ) depended on the assumption λ(±2β) < ∞. Now working under the more natural hypothesis (1.2), we must define a one-parameter family of metrics (d α ) α>1 , whereby α can be chosen so that αβ < β max . For each α > 1, it must be checked that d α is a metric, and that it induces a compact topology.
Second, the definition of T depends very explicitly on P , the law of the reference walk. When this walk is SRW, showing continuity of T amounts to proving that local interactions between endpoints are modified continuously with the addition of another monomer. When P is general, however, there can be interactions of endpoints arbitrarily far apart. Showing that these interactions do not spoil continuity complicates the proof of Proposition 3.4. For the same reason, technical details become more difficult in the proof of Proposition 2.4, which is the generalization of Theorem B.
While this paper is focused on polymers in Z d , the above program could be carried out on other countable, locally finite Cayley graphs. In this setting, the translation action on Z d is generalized by the group's natural action on itself. One case of interest is the infinite binary tree, considered as a subset of the canonical Cayley graph for the free group on two generators. An intriguing observation is that while analogs of Theorems 1.1 and 1.2 should go through, the "favorite region conjecture" mentioned in Section 1.3 will not. Indeed, when the binary tree having 2 n leaves is identified with [0, 1] having 2 n dyadic subintervals, the limiting endpoint distribution of a low-temperature tree polymer converges in law to a purely atomic measure [9], in analogy with Theorem 1.1. While this measure sometimes confines almost all of its mass to a very narrow interval-in analogy with Theorem 1.2-the low probability of this event prevents a stronger localization result. For more on the comparison of tree polymers versus lattice polymers, see [23,Chapter 4].

Variational formulas for free energy
As in other statistical mechanical models, there is great interest in computing the limiting free energy p(β) from Theorem A. For P having finite support (i.e. P (y, x) > 0 for only finitely many x), a series of papers due to Georgiou, Rassoul-Agha, Seppäläinen, and Yilmaz [54,55,31] provides two types of variational formulas for p(β), one using cocycles (additive functions on N × Z d ) and another of a Gibbs variational form (optimizing the balance of energy versus entropy). A third type of variational formula appeared in [10] for the SRW case, and is extended to the general case in this paper as (4.8). This formula arises naturally as an optimization over a functional order parameter-in this case, the law of the endpoint distribution-following a variant of the cavity method used to study spin glass systems, in particular the Sherrington-Kirkpatrick model [59,60,51]. As such, the variational formula (4.8) can be considered in analogy with the Parisi formula [58], which has recently been the object of intense study (e.g. see [48-50, 20, 52, 4-6, 35, 7, 21]).

Martingale phase transition and free energy asymptotics
It is not difficult to check that the normalized partition function W n := Z n /E(Z n ) forms a positive martingale with respect to F n . Therefore, the martingale convergence theorem guarantees the existence of a random variable W ∞ ≥ 0 such that W n → W ∞ almost surely, while Kolmogorov's 0-1 law implies P(W ∞ > 0) ∈ {0, 1}. It is known in the SRW case [29, Theorem 3.2(a)] and in the long-range case (stated but not proved in [66,Theorem 1.4]) that there is a phase transition. That is, there existsβ c ∈ [0, ∞] such that β <β c ⇒ P(W ∞ > 0) = 1 ("weak disorder") β >β c ⇒ P(W ∞ = 0) = 1 ("strong disorder").
Since W n → 0 exponentially quickly when β > β c , it is clear thatβ c ≤ β c , although it is conjectured thatβ c = β c in general. 2 This is known only in exceptional (but highly non-trivial) cases whenβ c = 0. For the SRW case, it was proved that β c = 0 by Comets and Vargas for d = 1 [ 3 Recently, Wei showed [65, Theorem 1.3] that in the critical case α = d = 1 and under some regularity assumptions on P ,β c = 0 again implies β c = 0.
The results just mentioned from [36,66,65] are in fact corollaries of asymptotics obtained for p(β) as β 0. In the SRW case, the bounds in [36] have been subsequently sharpened [64,47,3,11]. For d = 1, the precise exponent seen in these results is related to the scaling of β n 0 that generates the intermediate disorder regime in which a rescaled lattice polymer converges to the continuum random polymer [2]. This method of identifying a KPZ regime was initiated in [1] and extended to certain long-range cases in [16,17]. In fact, the authors of [17] are able to identify a universal limit for the point-to-point log partition functions, in critical cases, for both d = 1 and d = 2. Related work on the stochastic heat equation has been done for d ≥ 3 [45]. Finally, in [24] the asymptotics of p(β) as β → ±∞ are derived when P has a stretched exponential tail, and the environment consists of Bernoulli random variables.

Continuous versions of long-range polymers
In [44], a continuous version of α-stable long-range polymers is considered. Specifically, a phase transition was shown for the normalized partition function associated to a Lévy process subjected to a Poissonian random environment. Sufficient conditions were given for either side of the transition. In the same way that the α-stable polymer introduced in [22] generalized classical lattice polymers, this Lévy process model generalized a Brownian motion in Poissonian environment, which was introduced in [28] and also considered in [37,30].

Free energy and phase transition
In order to give context for the main results, which concern the behavior of polymer measures above and below a phase transition, we must first check that such a phase transition exists. To do so, we need to prove Theorems A and B in the general setting. First, in Section 2.1 we show that the quenched free energy has a deterministic limit. The arguments used here are standard, and the expert reader may skip them; nevertheless, the details are included to verify that no essential facts are lost when working with an arbitrary reference walk. Next, a proof of the phase transition is given in Section 2.2. In particular, the methods initiated in [29] must be refined to account for general P and weaker assumptions on the logarithmic moment generating function λ(β).

Convergence of free energy
We begin by showing the existence of a limiting free energy. Throughout this section one may assume a condition just slightly weaker than (1.2), namely λ(±β) < ∞.
The proof follows the usual program of showing first that E(F n ) converges, and second that F n concentrates around its mean.
The nonnegativity of all summands allows us, by Tonelli's theorem, to pass the expectation through the sum. That is, In particular, the inequality in (2.2) follows from the above display. On the other hand, a lower bound is found by again using Jensen's inequality, but applied to t → e t and with respect to P : where the final equality is a consequence of Fubini's theorem. Indeed, we may apply Fubini's theorem because Therefore, we obtain the desired lower bound: Now we may prove superadditivity. For a given integer k ≥ 0 and y ∈ Z d , let θ k,y be the associated time-space translation of the environment: , the random variables Z n and Z n • θ k,y have the same law. Furthermore, for any 0 ≤ k ≤ n, we have the identity where Z k (y) := E(e −βH k (ω) ; ω k = y) is the contribution to Z k coming from the endpoint y. By Jensen's inequality, Notice that Z n−k • θ k,y depends only on the environment after time k, meaning it is independent of F k . We thus have Using this observation in the previous display, we arrive at the desired superadditive inequality: EJP 23 (2018), paper 30.

Existence of critical temperature
Now we prove a phase transition between the high temperature and low temperature regimes. • Two generalizations were claimed to follow easily from the same methods. Vargas [62,Lemma 3.4] suggests the hypothesis β max = ∞ can be dropped, while Comets [22, Theorem 6.1] allows P to be α-stable, α ∈ [0, 2). We do both by assuming only β max > 0 and allowing P to be a general random walk. It seems the resulting difficulties are only technical, but how to resolve them is not obvious, and so we provide a full proof.
Proof of Proposition 2.4. Note that log Z n = E(e −βHn(ω) ) is the (random) logarithmic moment generating function for −H n (ω) with respect to P , and is finite for all β ∈ [0, β max ) almost surely by the proof of Lemma 2.2. Therefore, E(log Z n ) is convex on (0, β max ). Now seen to be the limit of convex functions, p must be convex on (0, β max ). It follows from general convex function theory that p is differentiable almost everywhere on (0, β max ), and Furthermore, convexity implies p is absolutely continuous on any closed subinterval of [0, β max ), and thus Then we could conclude that In particular, the existence of β c will be proved. Therefore, we need only to show (2.6). To do so, we would but each of (a), (b), and (c) require justification. Postponing these technical verifications for the moment, we complete the proof of (2.6) assuming (2.7).
For a fixed ω ∈ Ω p , we can write where E denotes expectation with respect to the probability measure P given by dP.
Since the Radon-Nikodym derivative is a product of independent quantities (with respect to P), the probability measure P remains a product measure. Therefore, we can apply the Harris-FKG inequality (e.g. see Therefore, where the penultimate equality is a consequence of Tonelli's theorem, since Z −1 n e −βHn(ω) > 0. The inequality (2.6) now follows by dividing by n. (2.7) Fix β ∈ (0, β max ). Choose q > 1 such that qβ < β max , and let q be its Hölder conjugate:

Justification of (c) in
Step (c) in (2.7) will follow from Fubini's theorem once we verify that Let g and h be as in Lemma 2.6, so that we can write Temporarily fix a path ω ∈ Ω p . Since Z n and −H n , and therefore g(−H n ), are nondecreasing functions of all η(i, x), the Harris-FKG inequality shows (2.11) The first factor satisfies The second factor satisfies where we have used Tonelli's theorem to exchange the order of integration. We have thus shown as desired.

Justification of (a) in
We will ultimately invoke dominated convergence to pull the limit through the expectation.
Consider any fixed h satisfying |h| ≤ ε. Now, log Z n is almost surely continuously differentiable, and so by the mean value theorem, a.s.
for some ε η depending on the random environment η and satisfying |ε η | ≤ |h| ≤ ε. But then by convexity of log Z n , we can bound the difference quotient by consider the endpoints of the interval [β 0 − ε, β 0 + ε]: where the maximum is now a dominating function independent of h. We showed (2.17) holds almost surely, and so < ∞.
In particular, the dominating function is integrable:

Adaptation of abstract machinery
In this section we recall and adapt some necessary definitions and results from [10]. The key change we will make is to the definition of the "update map" that sends a fixed endpoint distribution ρ n (ω n = ·) to the conditional law of ρ n+1 (ω n+1 = ·) given F n , viewed as a random variable in a suitable metric space (S, d). The construction of S is identical to what was done in [10]; we briefly describe it in Section 3.1 and justify some adaptations in Section 3.2. Then, in Section 3.3 we newly define the update map to allow for general reference walk P , and then lift it to a map of probability measures on S, a space denoted P(S). Finally, in Section 3.4 we prove continuity of the update map with respect to Wasserstein distance, which implies continuity of its lift.

Partitioned subprobability measures
The quenched endpoint distribution at time n, given by is a Borel measurable function of η when considered in the space 1 (Z d ).  For each α > 1, we construct a pseudometric d α on S 0 as follows. Define "addition" and "subtraction" on N × Z d by extending the group structure of Z d , but only if the first coordinates agree: Similarly, define the 1 "norm" by The maximum integer m for which (3.3) holds (possibly infinite) is called the maximum degree of φ, and is denoted deg(φ). The following lemmas demonstrate two useful properties of isometries: composition and extension.  . Suppose that φ : A → N × Z d is an isometry of degree m ≥ 3. Then φ can be extended to an isometry Φ : By induction, if φ has deg(φ) ≥ 2k + m, then φ can be extended to an isometry Φ : Given an isometry φ (which implicitly stands for the pair (A, φ)) and α > 1, we define the α-distance function according to φ: Finally, the pseudometric is obtained by taking the optimal α-distance: where deg(φ) ≥ 1 means φ is injective. The case α = 2 was considered in [10], and we can easily adapt the proof given there to show d α satisfies the triangle inequality. Since d α is clearly symmetric in f and g (by changing φ to φ −1 ), this result verifies that d α is a pseudometric.
With this pseudometric, a new space is realized by taking the quotient of S 0 with respect to d α : That is, S is the set of equivalence classes of S 0 under the equivalence relation We call S the space of partitioned subprobability measures, and it naturally inherits the metric d α . Lemma 3.4 below shows that for distinct α, α > 1, we have d α (f, g) = 0 if and only if d α (f, g) = 0. Therefore, we are justified in not decorating the space S with an α parameter, since S 0 /(d α = 0) is always the same set.
The quotient map ι : S 0 → S that sends an element to its equivalence class is Borel measurable with respect to the metric topology, cf. [ The support number of f is the cardinality of H f , which is possibly infinite. For g ∈ S 0 with N-support H g , d α (f, g) = 0 if and only if there is a bijection σ : (3.10) The key fact, and indeed the goal of constructing S, is the following result. It was proved for α = 2 in [10, Theorem 2.9], and once more the proof readily extends to any α > 1 by a modification as simple as changing the 2's to α's. We will generally write f for an element of S, and explicitly indicate f ∈ S 0 when referring to a representative in S 0 . When f is being evaluated at some u ∈ N × Z d , a representative has been chosen. For certain global functionals such as · defined in Here Π(µ, ν) denotes the set of probability measures on S ×S having µ and ν as marginals. Lemma 3.5 implies (P(S), W α ) is also a compact metric space. It is a standard fact (for instance, see [63, Theorem 6.9]) that Wasserstein distance metrizes the topology of weak convergence. In the compact setting, weak convergence is equivalent to convergence of continuous test functions. If L is (lower/upper semi-)continuous, then L is (lower/upper semi-)continuous.

Equivalence of generalized metrics
We have introduced a family of metrics (d α ) α>1 on S, where the flexibility of choosing α sufficiently close to 1 will allow us to make more effective use of the abstract methods in [10]. Namely, the only assumption we need is (1.2). It is important, however, that each metric induces the same topology. The next proposition verifies this fact. In particular, any functional on S that was proved in [10] to be continuous with respect to d 2 remains continuous under d α , α > 1. Proof. Since α and α are interchangeable in the claim, it suffices prove the "only if" direction. That is, we assume d α (f, f n ) → 0 as n → ∞. Fix representatives f, f n ∈ S 0 . Given ε > 0, set Then choose N sufficiently large that d α (f, f n ) < δ for all n ≥ N . In particular, for any such n, there is an isometry φ n : A n → N × Z d satisfying d α,φn (f, f n ) < δ. In particular, These four inequalities together show As ε > 0 is arbitrary, it follows that d α (f, f n ) → 0.

Generalized update map
Throughout the remainder of the manuscript, we fix β ∈ (0, β max ) according to (1.2), and we also fix some α > 1 such that αβ < β max . We then restrict our attention to S equipped with the metric d α , and P(S) with W α . Proposition 3.8 tells us that the topology on S does not depend on α, although the same is not true for the topology on P(S) induced by W α . Indeed, there can exist functions ϕ : S → R which are Lipschitz-1 with respect to some d α but not Lipschitz at all with respect to some other d α .
We write f n to denote the (random) endpoint distribution under the polymer measure ρ n , belonging to either 1 (Z d ) or S depending on context. Notice that we have the  .
This identity shows how f 0 → f 1 → · · · forms a Markov chain when embedded into S. Namely, we identify f n−1 with its equivalence class in S so that a representative takes values on N × Z d instead of Z d . (In this case, the support number is just 1.) Then the law of f n ∈ S given f = f n−1 is the law of the random variable F ∈ S defined by , (3.12) where (η w ) w∈N×Z d is an i.i.d. collection of random variables having the same law as η, and P (v, w) = P (y, z) if v = (n, y), w = (n, z) 0 otherwise.
To simplify notation, we write v ∼ w in the first case (i.e. v and w have the same first coordinate) and v w otherwise.

Remark 3.9.
Although the indexing of η w by w ∈ N × Z d might appear to reflect a notion of time, we are not using N to consider time. Rather, in order to compactify the space of measures on Z d , we needed to pass to subprobability measures on N × Z d . To avoid confusion, we will never write N to index time. Following this rule, we will write η w whenever we wish to think of a random environment on N × Z d , always at a fixed time.
When considering the original random environment defining the polymer measures, we will follow the standard η(i, x) notation. In either case, we will continue to use boldface η when referring to the entire collection of environment random variables.
Generalizing (3.12) to f ∈ S that may have f < 1, we define T f ∈ P(S) to be the law of F ∈ S defined by Notice that the expectation (with respect to η) of the numerator is e λ(β) v∼u f (v)P (v, u), while the expectation of the denominator is e λ(β) . Therefore, these quantities are almost surely finite, and so F is well-defined. In order for T f to be well-defined, we must check the following: (i) Given any f ∈ S 0 , the map R N×Z d → S given by η → F is Borel measurable, where R N×Z d is equipped with the product topology and product measure (L η ) ⊗N×Z d , and L η is the law of η. Claim (i) is immediate, since η → F is clearly a measurable map from R N×Z d to S 0 . After all, it is simply the quotient of sums of measurable functions. And then F → ι(F ) from S 0 to S is measurable by [10,Lemma 2.12]. Claim (ii) is given by the following lemma.   (3.14) where the ζ w are i.i.d., each having law L η . Then when these functions are mapped into S by ι, the law of F is equal to the law of G.
Proof. To show that F and G have the same law, it suffices to exhibit a coupling of the environments η and ζ such that F = G in S. So we let H f and H g denote the N-supports of f and g, respectively, and take σ : Next we couple the environments. Let ζ u be equal to η ψ −1 (u) whenever u ∈ H g × Z d . Otherwise, we may take ζ u to be an independent copy of η u . Now, for any u = (n, x) and v = (n, y) with n ∈ H f ,  Together, (3.17), (3.18), and the fact that f = g (cf. discussion following Lemma 3.5) Letting ε tend to 0 gives the desired result.
We have now verified that the map S → P(S) given by f → T f is well-defined. It remains to be seen that the map is measurable, although this fact will be implied by EJP 23 (2018), paper 30. the continuity proved in the next section. Given measurability, we can naturally lift the update map to a map on measures. For µ ∈ P(S), define the mixture T µ(dg) := T f (dg) µ(df ), (3.19) which means More generally, for all measurable functions ϕ : S → R that either are nonnegative or satisfy In this notation, we have a map P(S) → P(S) given by µ → T µ. We can recover the map T by restricting to Dirac measures; that is, T f = T δ f , where δ f ∈ P(S) is the unit point mass at f . For our purposes here, it suffices to know that µ → T µ is continuous, by an argument which requires only the continuity of f → T f (see [10, Appendix B.1]).

Continuity of update map
The goal of this section is to prove the following result. d α (f, g) < δ ⇒ W α (T f, T g) < ε.
The proof will proceed in a similar manner as in the nearest-neighbor random walk case, although modification is necessary to account for the fact that now the set of "neighbors" may be arbitrarily large, even all of Z d . In preparation, we record the following results.
Lemma 3.13. Let X 1 , X 2 , . . . be i.i.d. copies of an integrable, centered random variable X. For any t > 0, there exists b > 0 such that whenever c 1 , c 2 , . . . are constants satisfying (3.23) Given t and L, let b > 0 be sufficiently small that . (3.24) As in the hypothesis, assume |c i | ≤ b for all i, and (3.25) In order to apply martingale inequalities, we recenter the remaining sum: For each i, the random variable c i (X i − E(X)) has mean 0 and takes values between −|c i |(L + 1) and |c i |(L + 1). Therefore, by the Azuma-Hoeffding inequality [13, Theorem Proof of Proposition 3.11. Since (S, d α ) is a compact metric space, uniform continuity of f → T f will be implied by continuity. So it suffices to prove continuity at a fixed f ∈ S. Let ε > 0 be given. To prove continuity at f ∈ S, we need to exhibit δ > 0 such that if g ∈ S satisfies d α (f, g) < δ, then there exist representatives f, g ∈ S 0 and a coupling of environments (η, ζ) such that under this coupling the following inequality holds: Here F, G ∈ S are given by (3.13) and (3.14). We shall see that it suffices to choose δ satisfying conditions (3.32a)-(3.32d) below.
Let q := max x∈Z d P (0, x), which is strictly less than 1 by (1.1). Next choose t > 0 sufficiently small to satisfy one of the following conditions:  κ ≤ 1 7 Then fix a representative f ∈ S 0 , and choose A ⊂ N × Z d finite but sufficiently large that By possibly omitting some elements, we may assume f is strictly positive on A. Next, let k be a positive integer sufficiently large that If A is nonempty, we will also demand that (2d) k |A|δ 1/α < κ. (3.32b) Otherwise, we relax this assumption to δ 1/α < κ. (3.32c) Finally, we assume We claim this δ > 0 is sufficient for ε > 0 in the sense described above. Assume g ∈ S satisfies d α (f, g) < δ. Then given any representative g ∈ S 0 , there exists an isometry ψ : By (3.32a), it follows that deg(ψ) > 3k. In particular, upon defining φ := ψ| A , we have deg(φ) ≥ deg(ψ) > 3k. Therefore, Lemma 3.2 guarantees that φ can be extended to an isometry Φ : Alternatively, we can express this set as The inequality deg(Φ) > k implies that for any B ⊂ A, Furthermore, (3.33) and (3.32d) together force A ⊂ C, since Given a random environment η, we couple to it ζ in the following way. If u ∈ Φ(A (k) ), then set ζ u = η Φ −1 (u) . Otherwise, let ζ u be an independent copy of η u . Note that F and G are now distributed as T f and T g, respectively, when mapped to S. On the other hand, Φ is deterministic, and so no measurability issues arise in the bound It thus suffices to show E(d α,Φ (F, G)) < ε. To simplify notation, we will write so that F (u) = f (u)/ F and G(u) = g(u)/ G.
Consider the first of the four terms on the right-hand side of (3.35). Observe that for Summing over A (k) and taking expectation, we obtain the following from the first term in the last line of (3.36): Meanwhile, the second term in (3.36) gives These preliminary steps suggest two quantities we should control from above, namely the right-hand sides of (3.37) and (3.38). Consideration of the second and third terms EJP 23 (2018), paper 30. on the right-hand side of (3.35) suggests four more quantities, since the Harris-FKG inequality yields and similarly Therefore, we should seek an upper bound for E( F −α ) and E( G −α ), as well as E u / For clarity of presentation, we divide our task into the next four subsections.

Upper bound for
On the other hand, and so if f > 0, then Lemma 3.12 gives

Upper bound for
, meaning | f (u) − g(Φ(u))| is a non-decreasing function of η u and independent of η w for w = u. Since F −1 is a non-increasing function of all η w , the Harris-FKG inequality yields where Of the two absolute values above, the second is easier to control. Indeed, Next consider the first absolute value in the final line of (3.46). The difference between the two sums can be bounded as Each of the four terms above can be controlled separately. For the first term, notice that That is, if ψ(v) ∈ N (Φ(u), k) ∩ ψ(A), then v ∈ N (u, k) ∩ A and satisfies P (v, u) = P (ω 1 = u − v) = P (ω 1 = Φ(u) − ψ(v)) = P (ψ(v), Φ(u)). We thus have a bijection between N (u, k) ∩ A and N (Φ(u), k) ∩ ψ(A), meaning where the final inequality is trivial since A ⊂ C. Summing over u ∈ A (k) gives the total Considering the second term on the right-hand side of (3.48), we have (3.30) < κ. (3.50) Similarly, for the third term, Finally, for the fourth term, 33) < (2d) k |A|δ 1/α (3.32b) < κ. Combining (3.48)-(3.52), we arrive at (3.53) Using (3.47) and (3.53) with (3.46) reveals and thus we have the desired bound: (3.54)
Therefore, by the choice of b in relation to Lemma 3.13, we deduce from (3.56) that E| F − G| ≤ 3t. (3.60) There are now two cases to consider. First, if f < 1, then On the other hand, if f = 1, then we can consider the three sums in (3.56) separately. For any u ∈ A (k) , the quantity We must also have and so by applying Lemma 3.12 once more, we see Again appealing to independence and then Lemma 3.13, we obtain the bound Finally, we use independence and Lemma 3.13 once more to obtain Combining (3.62)-(3.64), we conclude that if f = 1, then   where the first sum is bounded by 5κ according to (3.53), and the second sum satisfies Hence (3.57) holds: Next consider the c u . By definition of A (k) , Last, consider the c u . We have the implication and thus (3.30) < d α,ψ (f, g) + κ (3.33) < δ + κ

Variational formula for free energy
Given the results of Sections 2 and 3, there is little difficulty left in obtaining the main results. Indeed, the majority of remaining proofs go through as in [10] with no change. The main reason for this is the high level of abstraction in our approach, the essential components of which are S and T . Since T can be thought of as a continuous transition kernel for a Markov chain in a compact space S, general ergodic theory provides swift access to results on Cesàro limits. Furthermore, the high and low temperature phases can be characterized using the variational formula (4.8) of Theorem 4.1.

Outline of abstract methods
Let us now recall a progression of results, with appropriate references to [10]. Whenever modification is necessary, an updated proof is provided in Section 4.2. Recall from (3.1) that f n is the quenched endpoint distribution of the polymer at length n, identified as an element of S.  which is a random element of P(S), measurable with respect to F n . Since T f n is the law of f n+1 given F n , a martingale argument shows that µ n almost surely converges to the set of fixed points of T , which we denote K := {ν ∈ P(S) : T ν = ν}.   When f = f n is the n-th endpoint distribution, (3.11) shows Therefore, Upon lifting R to the map R : P(S) → R given by we obtain a continuous functional on measures by Lemma 3.7. Furthermore, the above calculation can be rewritten as E(R(µ n−1 )) = E(F n ). It is proved in [10,Proposition 4.6] that in fact R(µ n−1 ) − F n → 0 almost surely as n → ∞, and so Proposition 2.1 implies The only part of the proof that requires modification is stated as Lemma 4.2 in the next section.

Proofs in general setting
Once the lemmas of this section are checked, all of the results stated in Section 4.1 will be proved.

Adaptation of a fourth moment bound
The proof of [10, Proposition 4.6] uses the Burkholder-Davis-Gundy (BDG) inequality [15, Theorem 1.1] applied to a martingale whose differences are of form W − E(W ), where W is a random variable defined using a fixed f ∈ S 0 with f = 1. Specifically, thus making the BDG inequality useful, but the proof assumed λ(±2β) < ∞. Here we make a simple adaptation of the proof that assumes only λ(±β) < ∞ and obtains a different value for C.

Adaptation of a free energy inequality
The proof of (4.8) can be written exactly as the proof of Theorem 4.9 in [10], but it requires the result of the next lemma. We introduce the boldface notation 1 to denote the element of S having representatives in S 0 of the form Similarly, 0 will denote the element of S whose unique representative is the constant zero function.

Lemma 4.3.
For any f 0 ∈ S and n ≥ 1, where δ f0 ∈ P(S) is the unit mass at f 0 . Equality holds if and only if f 0 = 1.
Proof. Fix a representative f 0 ∈ S 0 . Let (η (i) u ) u∈N×Z d , 1 ≤ i ≤ n, be independent collections of i.i.d. random variables with law L η . For 1 ≤ i ≤ n, inductively define f i ∈ S to have representative   Observe that when i = n, the first summand in (4.12) is equal to un∈N×Z d un−1∼un f n−1 (u n−1 ) exp(βη (n) un )P (u n−1 , u n ) = 1 D n−1 un∈N×Z d un−2∼un−1∼un f n−2 (u n−2 ) exp(βη (n−1) un−1 + βη (n) un )P (u n−2 , u n−1 )P (u n−1 , u n ) . . .  By summing the final expressions in (4.14) and (4.15) to obtain the right-hand side of (4.12), and then clearing the fraction, we see Using the concavity of the log function, we further deduce where equality holds throughout if and only if f 0 (u 0 ) = 1 for some u 0 ∈ Z d . Since the random variable un∼un−1∼···∼u0 exp β n i=1 η (i) ui n i=1 P (u i−1 , u i ) is equal in law to Z n EJP 23 (2018), paper 30.
It follows that with equality if and only if f 0 = 1.
Since (S, d α ) has an equivalent topology to (S, d 2 ) by Proposition 3.8, continuous functionals on the latter space are also continuous on the former. Consequently, the proof in [10, Section 6] given for the SRW case requires no modification.

Geometric localization with positive density
As usual, let f i (·) = ρ i (ω i = ·) denote the probability mass function for the i-th endpoint distribution. We say that the sequence (f i ) i≥0 exhibits geometric localization with positive density if for every δ > 0, there is K < ∞ and θ > 0 such that lim inf   As for Theorem 5.3, the proof is equivalent to the one in the SRW case, which the reader can find in [10,Section 7].