Infinite disorder renormalization fixed point for the continuum random field Ising chain

We consider the continuum version of the random field Ising model in one dimension: this model arises naturally as weak disorder scaling limit of the original Ising model. Like for the Ising model, a spin configuration is conveniently described as a sequence of spin domains with alternating signs (domain-wall structure). We show that for fixed centered external field and as spin-spin couplings become large, the domain-wall structure scales to a disorder dependent limit that coincides with the infinite disorder fixed point process introduced by D. S. Fisher in the context of zero temperature quantum Ising chains. In particular, our results establish a number of predictions that one can find in [Fisher, Le Doussal and Monthus 2001]. The infinite disorder fixed point process for centered external field is equivalently described in terms of the process of suitably selected extrema of a Brownian trajectory introduced and studied by J. Neveu and J. Pitman [Neveu and Pitman 1989]. This characterization of the infinite disorder fixed point is one of the important ingredients of our analysis.


Introduction
Understanding the behavior of disordered systems is often extremely challenging, above all when the disorder modifies in a substantial way the behavior of the system with respect to when disorder is absent (pure model ).One of the most basic and highly studied models in this field is the ferromagnetic Random Field Ising Chain (RFIC), that is the probability measure on {−1, 1} V N , V N := {1, 2, . . ., N }, with (discrete) probability density proportional to exp for σ ∈ {−1, 1} V N and for a choice of the boundary condition σ 0 .In addition, J ≥ 0 is the ferromagnetic interaction that plays in favor of alignment between spins, and the real values h 1 , h 2 , . . .form the sequence of external fields: for us they will be (the realization of) a sequence of IID random variables with E[exp(th)] < ∞ for t in a neighborhood of the origin.In the end this work is only about E[h] = 0, even if at some point, mostly for presentation issues, the case E[h] ̸ = 0 will be considered too.
It is well known that this model does not exhibit a phase transition, so there is no critical phenomenon.However, a pseudo-critical behavior appears in the J → ∞ limit (we always consider N → ∞ before J → ∞).We aim at understanding the configurations of the system in this limit: • In the pure case, i.e. h j = 0 for every j, the model is equivalent to a Markov chain with state space {−1, +1} with transition probability from +1 (respectively, −1) to −1 (respectively, +1) equal to 1/(1 + e 2J ).Therefore the typical configurations for J large are very long domains of aligned spins: the distance between the walls (i.e., the spin flip locations that identify the interfaces between domains of different sign) are independent geometric variables with mean 1 + e 2J .• As soon as disorder is present the model is no longer solvable, but an Imry-Ma argument (see for example [26, p. 373]) rapidly leads to guessing that the domain-wall structure changes radically and that the expected spatial scale drastically reduces to J 2 /Var(h).
The Imry-Ma argument may be given as follows: if we are in a (say) −1 domain region, switching to +1 in an interval I L of length L costs two spin flips (i.e., 4J) but the energy change due to the external fields is 2 j∈I L h j which, for L large, is 2 L Var(h) times a Gaussian standard variable.Hence, if L ∝ J 2 /Var(h), by choosing wisely (and in a disorder dependent fashion!)I L , the energetic gain from the external random field can overcome the spin flip cost.This not only strongly suggests that J 2 /Var(h) is the correct space scale, but also that the wall positions heavily depend on the disorder configuration (h j ).
The Imry-Ma argument we just outlined therefore suggests that disorder is strongly relevant for the RFIC in the J → ∞ limit.This terminology -at least the term relevant -is not only suggestive of the fact that disorder changes the behavior of the system, and in a substantial way: it is a standard terminology in the Renormalization Group (RG) context (see for example [23, § 5.3] and [26,42]).Disorder is dubbed relevant when it changes the RG fixed point (in practice, the large scale behavior of the system) with respect to the pure system: as we are going to explain, in the case we are considering the disorder ends up having an infinitely strong effect.To make a long story short, it is in the RG community that a very precise description of the domain-wall structure has been set forth.In fact, in [21] D. S. Fisher, P. Le Doussal and C. Monthus claim that the J → ∞ domain-wall structure is sharply captured by the infinite disorder fixed point identified by D. S. Fisher in the framework of disordered quantum chains [19,20].
It should be noted that in [21] the equilibrium case is considered as well as the problem of the approach to equilibrium: for the equilibrium case a vast literature is cited (we single out the extensive review monograph [34]), but the precise description of the domain-wall structure, to our knowledge, first appears in [21].We only consider the equilibrium model.
Fisher's infinite disorder fixed point and, more generally, Fisher's approach is expected to apply to a wide universality class of models.We refer to the review papers [26,27,42] for extended accounts, but we stress that how wide this universality class is represents a very challenging problem, and not only from a mathematical viewpoint (for example, in [9,17] a special pinning model is shown not to belong to this class, suggesting that general pinning models [23], see [27] for the RG approach, do not belong either).Nevertheless it is worth pointing out that the most basic model to which Fisher's ideas apply is Random Walk in Random Environment (RWRE).The mathematical understanding of RWRE, above all in one dimension, is extremely advanced (see [43]) and the agreement with Fisher's RG approach has been verified at least in part.We are not going into a detailed discussion of this issue and we refer to [8,4].We rather point to the fact that in [8,4], see also [18], it is spelled out that Fisher's infinite disorder fixed point is a stochastic process that was already known in the mathematical literature: it is the Brownian motion Γextrema process introduced and studied by J. Pitman and J. Neveu in [36].In Fisher's approach instead, the Γ-extrema process arises from a dynamical coarse graining procedure that progressively absorbes the small spin domains into the neighboring ones: the coarse graining procedure is stopped when all the domains have size Γ or more (Γ = 2J in our case).We insist on the fact that Fisher's fixed point, alias Pitman-Neveu process, just depends on the disorder (that is, on an IID sequence that, in the large scale limit, becomes a white noise or, equivalently, a Brownian motion).We stress that we have kept the original presentation of the Pitman-Neveu process, which is rather a family of Γ-index processes.But, as spelled out in Remark 2.1, the Pitman-Neveu process enjoys a scale invariance and Fisher's fixed point can be seen, in a more customary fashion, as the Pitman-Neveu process with Γ = 1.Two sources of randomness are present in a disordered system: the disorder itself, and the thermal fluctuations of the spins.Characterizing the disorder as infinite in Fisher RG fixed point is directly linked to the fact that disorder (drastically!)dominates over thermal fluctuations whenever Fisher's approach applies.
The purpose of this work is to give a precise statement that explains how Fisher's infinite disorder fixed point captures the domain-wall structure in a continuum version of the one-dimensional Ising chain with centered random external field.The continuum version of the Ising model we consider appears in the literature since the end of the 60s [35] where the authors analyze the weak disorder scaling limit for the two dimensional Ising model with interaction disorder of columnar type [35,11]: the key point is that the two dimensional partition function is written in terms of a family of (one dimensional) RFIC transfer-matrices.We signal that the naturally related, but very different issue of the weak disorder scaling limit of the two dimensional Ising model with random external field is considered in [5].
Weak disorder scaling limits were first introduced in [22] in a different context (Anderson localization in one dimension).For a review of the literature on this limit we refer to [11,12] and references therein.Much more generally and recently, weak disorder scaling limits have attracted a lot of attention, both because of the mathematicians' efforts to understand disorder relevance in statistical mechanics models and because of their link with singular SPDEs [1,7].We stress that, in spite of the fact that these continuum models arise as weak disorder scaling limit of the original models, they appear to capture phenomena in which disorder strongly dominates on large scales.And this is in fact what we prove for the model we consider: we show that to leading order the behavior of the model is fully captured by Fisher's infinite disorder fixed point.

The model and the main results
2.1.The Continuum RFIC.(N t ) t≥0 denotes a Poisson process with intensity (2.1) We denote by P Γ the law of (N t ) t≥0 and we introduce the free spin process s t := (−1) Nt ∈ {−1, 1} for t ≥ 0. So, under P 0 , (N t ) t≥0 is a Poisson process of rate exp(−0) = 1.We follow the standard convention of choosing the right-continuous version of (N t ) t≥0 .P denotes the law of B = (B t ) t≥0 , which is a standard Brownian motion.The partition function of the Continuum Random Field Ising Chain (Continuum RFIC) of length ℓ > 0 is with α ∈ R an asymmetry parameter that we introduced for presentation purposes -it corresponds to E[h] in the (discrete) RFIC -and the results in this work just concern the case α = 0: so, unless explicitly specified otherwise, α = 0. Note that in the standard RFIC (1.1) we can write the energy due to the nearest-neighbor interaction as JN minus Γ(= 2J) times the number of nearest-neighbor spins that have different sign: hence the second line in (2.2) makes clear the link between RFIC and Continuum RFIC (see Appendix C for a deeper analysis of this link).The partition function (2.2) has the superscript f because the boundary condition on the right is free (and on the left it is +1: note that s 0 = +1 since N 0 = 0).But we will mostly work with That is, the boundary conditions are +1 on both ends: we use the standard notation with X a random variable and A an event.There is no fundamental reason to choose these boundary conditions and in fact the results we prove hold with any choice of boundary conditions ±1 on the right and on the left (see Remark 5.9).Hence the results hold also for the free model corresponding to (2.2).But for definiteness, and without loss of generality, we focus on (2.3).
We denote by µ Γ,B•,ℓ the Gibbs measure associated with Z Γ,B•,ℓ , in particular for t ∈ By construction, this Gibbs measure gives probability one to the right-continuous configurations s • taking values in {−1, +1} and such that s 0 = s ℓ = +1.
2.2.Neveu-Pitman Γ-extrema and Fisher's RG fixed point.To present Fisher's RG fixed point and our result we need some notation.For the next paragraph it is useful to refer to Fig. 1.
We then introduce the Fisher trajectory s (F )

•
= (s (F ) t ) t≥0 .For t ≥ 0 we set where for this definition we have introduced u 0 (Γ) = 0.In an informal way, s (F ) t = +1 (respectively −1) if t is in an ascending stretch (respectively, in a descending stretch) and s (F,R) t = 0 if there is a Γ-extremum at t. Remark 2.1.Note that the t ↑ n (Γ) and t ↓ n (Γ) times are stopping times, but the u n (Γ)'s are not stopping times.Nevertheless they generate a remarkable renewal structure and relevant to us is notably that .. is an IID sequence of random vectors in (0, ∞) 2 whose distribution is explicitly known [36].In particular, both entries of the vector have exponentially decaying probability tails and the expectation is (Γ 2 , 2Γ) and the following scaling invariance holds: for every (2.12) The claim in [21] is that, in the limit J → ∞, the spin configuration is fully captured by s (F )  • .The computations are performed by assuming precisely s • , but this is clearly impossible because the thermal fluctuations induce wall fluctuations, i.e. fluctuations of the loci where the spin domains switch signs, of at least O(1).The authors of [21] are aware of this fact and they typically do not push their claims beyond what is expected to hold true (and, as we will explain in Section 2.4, for this model one can compute exactly the free energy for every J: this can of course be exploited in a number of ways, in particular as a sanity check).
A formula for D Γ appears in (2.32).
In order to properly discuss Theorem 2.2, in particular explaining the relations with the existing literature and exposing the open issues that it raises, we introduce some of the mathematical tools we exploit.We just anticipate that lim inf Γ→∞ ΓD Γ ≥ 1/2 (see Remark 2.7).The reason to prefer this setting is because it permits to explain (and exploit) in a more natural way a symmetry that is present in the system and that we present now.It will be practical to work also with η t := (1 − s t )/2 ∈ {0, 1}.In particular and by decomposing the denominator according to whether {η t = 0} or {η t = 1} we see that where we introduced and for every t < b. (2.21) We note that in (2.20), P Γ (η a = 0) = 1.There is a symmetry between L (a) • and R (b) • that is somewhat hidden by the expressions (2.20) and (2.21) and we present it here by pointing out that (see Section 5): • is the unique strong solution of the SDE driven by the bilateral Brownian motion for t ≥ a, initial condition L a = ∞ and we recall that ε = e −Γ .
−t ) t≥−b solves the same SDE, with the very same initial conditions, but with B and we solve this equation for t ≥ −b with R −b = ∞.
While a priori the fact that L a = R −b = ∞ may look like a source of problems and this issue is treated in Sec. 5, we anticipate that L t = 1/L t solves the same SDE, but with B t replaced by −B t and (above all) initial condition 0. Therefore L • falls into the standard existence and uniqueness framework.−t ) t≥−ℓ have the same law.
It is not difficult to show (see [11]) that the SDE (2.22) (and of course (2.23)) is reversible with respect to its unique invariant density: this is more practically presented in terms of l (a)   t := log L (a)  t and r (b) for which we have dl (a) ) with the strongly confining potential U Γ (x) := ε(exp(x) + exp(−x)).Therefore the density of the unique invariant probability p Γ of (2.25) is where K 0 (ε) is the modified Bessel function of second kind with index 0 evaluted in ε = e −Γ (given explicitly in (2.39), see [38,Ch. 10]), in particular 2K 0 (ε) = 2Γ + 2(log 2 − γ EM )+O(ε 2 log Γ) for Γ → ∞, where γ EM is the Euler-Mascheroni constant.Therefore this density is close for Γ large to the indicator function of (−Γ, Γ), divided by the normalization 2Γ.Note that l (a) • is nearly a Brownian motion (times 2) as long as it keeps away from ± log ε = ∓Γ.When it approaches ±Γ and, even more, when it tries to go far from (−Γ, Γ), a strong recalling force acts on it.Analogous observations of course apply to r (b)  • .It is also useful to introduce m (a,b) t := l (a) t + r (b) t and, with this notation, (2.19) has the particularly compact expression Without surprise, see [11], the law of l (a)  t (respectively r (b) t ) converges for t → ∞ (respectively, t → −∞) to p Γ .In Section 5 we are going to show the following related result: Lemma 2.4.For every t ∈ R and for almost every B • we have that the almost sure limits lim a→−∞ l (a) t =: l t and lim b→∞ r (b)  t =: r t exist.
Lemma 2.4 and several related results are included in Proposition 5.8: in particular the fact that the law of l t and of r t is p Γ for every t.Moreover (l t ) t∈R , respectively (r t ) t∈R , r t := r −t , can be characterized as the only strong solution for t ∈ R of Now we are going to state another fundamental result (proven in Section 6).For this result we need to introduce also the sequence of Γ-extrema of a bilateral Brownian motion.At this stage we do this informally: Appendix B is fully dedicated to this issue.But the key point is that by defining the Γ-extrema times starting from time 0, as we did, we create a sequence (u n (Γ)) n=1,2,... in which u 1 (Γ) is different in nature from u 2 (Γ), u 3 (Γ), . . .because the latter ones have an ascending Γ-stretch on one side and a descending one on the other.Instead, u 1 (Γ) has a Γ-stretch only on the right.If we repeat the procedure starting from an arbitrary time a < 0, instead of 0, the time u 1 (Γ) found starting from 0 may not be in the sequence of Γ-extrema that one finds starting from a. Nevertheless, u 2 (Γ), u 3 (Γ), . . .are Γ-extrema times in (u 1 (Γ), ∞) even if we define them starting from a < 0. This stability allows to define the full sequence (u (R)  n (Γ)) n∈Z (the indexation is chosen by requiring that u (R) 0 (Γ) < 0 and u (R) 1 (Γ) ≥ 0) of the Γ-extrema of a bilateral Brownian motion (see Appendix B for details).The point is that the sequence of the Γextrema of a bilateral Brownian motion partitions R into ascending Γ-stretches, descending Γ-stretches and the discrete set of Γ extrema.Therefore we can define (s (F,R) t ) t∈R , the Fisher trajectory for the bilateral Brownian motion: for every t ∈ R, s (F,R) t = +1 (respectively −1) it t is in an ascending (respectively descending) stretch, and s (F,R) t = 0 if there is a Γ-extremum at t.In a formula we obtain that implicitly defines (a (R) n ) n∈Z .For the next result, which is a reformulation of the first part of Theorem 2.2 with a more explicit expression for the constant D Γ , we actually need only s (F,R) 0 and part of the arguments in Appendix B are about recovering s (F,R) 0 without building the whole sequence.Proposition 2.5.For every Γ > 0 the following limit exists (2.32) Furthermore, for every Γ > 0 and for almost every B • , and, even more, in µ Γ,B•,ℓ -probability, (2.34) Proposition 2.5 is an ergodic type statement that is intuitive in the light of the identities in (2.27), so that, for example in order to establish (2.33), we are interested in the ℓ → ∞ limit of 1 ℓ t ), by Dominated Convergence and by applying the argument also to B (t)  • -defined in (2.16) -we have that for every t (2.36)This provides a pointwise result that complements Theorem 2.2: it says for example that We remark also that l 0 and r 0 are independent, because l 0 depends only on (B t ) t≤0 and r 0 depends only on (B t ) t≥0 .On the other hand, l 0 and r 0 have the same law p Γ .However this is not of much help to evaluate D Γ .Rather, we will aim at making the B • dependence of l 0 and r 0 explicit, so that we will understand the joint behavior of the random variables m 0 and s (F,R) 0 and, in turn, the behavior of D Γ .We complete this section by remarking that, by the expression for D Γ in (2.32), showing that D Γ tends to zero amounts to showing that (for Γ → ∞ and with high probability) m 0 is very large and positive (respectively negative) when s (F,R) 0 = +1 (respectively, s (F,R) 0 = −1).This is handled in Section 4.
2.4.About the free energy and the infinite disorder RG fixed point.A remarkable fact for this model is that the free energy density f α (Γ) is explicit [35,34,12,11].The free energy density is defined up to an additive constant: if we multiply the partition function by a factor that does not depend on s • , but which may depend on the parameters of the system (Γ, the system size ℓ, the disorder B • ) the Gibbs measure, i.e. the model itself, is not modified.In this sense and just for the sake of this discussion we introduce and analogous formula for (2.3).We recall that, under P 0 , η • is the standard Poisson process of rate exp(−0) = 1, see (2.2), and one directly checks that . So we define the free energy density f α (Γ) to be the exponential rate of growth of Z f Γ,B•,ℓ .More precisely we have that (a.s. and in L 1 , see [11]) where is the modified Bessel function of second kind of index α evaluated in x ∈ (0, ∞).This can be seen in various ways, in particular with the formalism we have introduced it is not difficult to see that for α = 0 a.s.
where for the first equality see Remark 5.3 and for the second one we have used that the law of l (0) t converges to p Γ when t → ∞ [11].By using the explicit expression for p Γ , given in (2.26), we recover (2.38) for α = 0.Moreover, by exploiting the known Bessel asymptotic behaviors [38,11], we see that where γ EM = 0, 5772 . . . is the Euler-Mascheroni constant.We refer to [25] for the analogous result for (discrete) RFIC.What we just presented, see in particular (2.40), shows that, in order to compute the free energy density, we just need the one-sided process l • and, ultimately, the key is given by the invariant probability p Γ of l • .If we want to get more information on the system we can differentiate the free energy with respect to the available parameters: for example for almost every B where we recall that N ℓ is the total number of Poisson events on [0, ℓ], so that (2.42) yields the density of domain walls.Two comments are in order: (1) The asymptotic behavior in (2.41) follows once again by known asymptotic behaviors [38, (10.25.2) and (10.27.4)].And one can push this computation further: notably, by taking one more derivative in Γ one obtains that the variance, under µ Γ,B•,ℓ , of N ℓ / √ ℓ converges for ℓ → ∞ a.s..The limit is explicit in terms of Bessel functions.Therefore, the wall density concentrates around its mean value as ℓ tends to ∞.
(2) A scaling argument yields that the free energy density for the model with . Hence, by differentiating with respect to λ, one obtains the density of disorder energy Remark 2.7.A well-known integration by part argument frequently used in the statistical mechanics of disordered systems, see Lemma C.2, shows that, when disorder is Gaussian (like in our case), the disorder energy density is directly related to the overlap.In our model the result is that ( where s (1)  • , s (2)  • denote two independent copies of s • under µ ⊗2 ℓ .An interesting implication of (2.44) follows by observing that s (1)  t ̸ = s (2)  t implies that s . Therefore, recalling (2.32), we have

.45)
As we have already pointed out, our main result (Theorem 2.2) is a quantitative and mathematically rigorous version of the fact that D. Fisher's infinite disorder fixed point does capture the leading order behavior of the domain-wall structure in the Γ → ∞ limit.This result does not allow to recover the extremely sharp control on some observables that one can extract from the explicit formula for the free energy of the Continuous RFIC, but it is really different in nature: (1) Via the explicit knowledge of the free energy density of the Continuum RFIC, see (2.38) and (2.41), the relevant observables can be computed explicitly in this model (see for example (2.42)).In [21] these observables are computed beyond leading order in Γ → ∞ using the infinite disorder fixed point and, by comparison with the exact results, one readily sees that the validity of the computations based on the infinite disorder fixed point does not go beyond leading order because of the thermal noise, which forces a positive discrepancy density D Γ , with the lower bound in Remark 2.7.This point is taken up again in Section 2.5.
(2) But one cannot infer, or even guess, the pathwise behavior from the knowledge of the free energy density.The pathwise behavior must of course be compatible on large scale with the correct behavior of the observables computed by differentiating the free energy density, but the prediction of the pathwise behavior comes from D. Fisher's RG procedure.The RG procedure is highly nontrivial and does not depend at all of the explicit, model dependent, free energy computations.
Remark 2.8.For a quantitative comparison of our results with the ones in [21] one should keep in mind that the constant Γ in [21] is twice our Γ.The value of Γ chosen in [21] is the interaction energy paid to generate a new domain, i.e. two walls.But, for example, the domain density [21, (45) and ( 39)] is 1/(2J) 2 (times the variance of the disorder), as in our case.

Open and related problems.
About more complete results for the Continuum RFIC.Our main result, Theorem 2.2, controls the proximity of s • and s (F ) • in a global sense and in probability (see Remark 2.6 for a pointwise result, always in probability).In Remark 4.3 we will explain that our results say that the process sign(m • ), m • defined in (2.30), is a better approximation of s • than s (F )   • is.On the other hand, s (F )   • is related in a much more elementary and intuitive way to the Brownian trajectory than sign(m • ).
But of course s (F ) • and sign(m • ) are just functions of the disorder, while thermal noise is present in the system, that is in s • .As a matter of fact, as discussed in Section 2.4, the infinite disorder RG fixed point s (F )   • captures the behavior of the free energy only up to the order 1/Γ (and similar considerations for other quantities of interest like the density of domain walls or the density of energy).It is not difficult to argue that the precise location of the domain walls fluctuates on a spatial scale of order 1 and, since the density of the domain walls is 1/Γ 2 , these (thermal) fluctuations lead to a correction of order 1/Γ 2 to the free energy density and to the density of energy.But thermal noise may also affect domains that are borderline from the energetic viewpoint.In fact, the domains we observe are the ones determined by the Γ-extrema and the absolute value of the energy (due to the disorder!) in a domain (i.e., the difference between the value of B • at two neighboring Γ-extrema) is at least Γ: if this energy differs from Γ just by a constant, the thermal noise may generate a global spin flip in the domain.It is not difficult to argue that such domain switch phenomena contribute to the free energy a term which is o(1/Γ 2 ), but they are contributing in a more substantial O(1/Γ) way to the overlap.
With respect to this issue, (2.41) (more precisely, just the second order expansion f 0 (Γ) = 1/Γ−(log 2−γ EM )/Γ 2 +o(1/Γ 2 )) strongly suggests (via a non trivial computation that we do not reproduce here) that the domain-wall boundaries actually should behave in the Γ → ∞ limit as diffusions in a (quenched) random potential given by a bilateral three-dimensional Bessel process.Establishing such a result is well beyond what we prove here, but the considerations we just developed lead to conjecturing that a finer approximation of the spin configurations is obtained, starting from the Fisher configuration, by randomizing the location of the domain wall locations (according to independent diffusions in a 3d Bessel environment), with the addition of another mechanism that randomly suppresses the domains whose disorder energy content is close to Γ.
Another direction in which Theorem 2.2 could be improved, notably in view of Theorem 3.3 for the simplified model (to which we hinted to in the caption of Fig. 2 and to which we dedicate Section 3), is understanding the behavior of D Γ in a sharper way.We do not know whether log •2 Γ can be replaced by a constant.
About the (discrete) RFIC.After the completion of this work we have been able to establish [10] a result similar to Theorem 2.2 for the (discrete) RFIC.The central tool in the discrete setting is the transfer-matrix method that in the disordered systems context, see for example [16,37,24,25], leads to considering random matrix products and the corresponding dynamics induced on a naturally associated projective space [41].We refer to [10] for details and discussion about the relation with the present work.
About the α ̸ = 0 case.D. Fisher, P. Le Doussal and C. Monthus in [21] deal also with the case α ̸ = 0 (that we use here as a synonymous of E[h] ̸ = 0).And the infinite disorder RG fixed point in the non centered case is considered in a number of other occasions, starting with the original works [19,20], see [26,42,23] and references therein.Strictly speaking, the limit process, or RG fixed point, for α ̸ = 0 is not the Neveu-Pitman process.But it is the process that describes the Γ-extrema (in terms of distances and height differences) of the Brownian motion with constant drift α: in this sense, it is the direct generalization of the Neveu-Pitman process (and it has been studied mathematically in [18]).Also in this case one is dealing with a process of independent random variables in [0, ∞) 2 , but they are not IID, rather the sequence of the variables with even (respectively odd) indexes are IID.This is simply due to the fact that (let us suppose that α > 0) ascending stretches are favored over the descending ones.The infinite disorder RG fixed point is explicit also for α ̸ = 0, but, unlike for the α = 0 case, it is clear that for α ̸ = 0 the infinite disorder RG fixed point gives just an approximate (or qualitative) picture of what really happens (mathematical arguments can be set forth in this direction, but this is implicitly admitted also in the physical literature [26,42] where the α = 0 case is considered to be exact and the α ̸ = 0 case is claimed to be only asymptotically exact, possibly referring to α → 0).The RG claims for α ̸ = 0 cannot hold because they do not agree with the precise asymptotic behavior of the free energy density: these can be found in [16,24] for the RFIC with α ̸ = 0 and the free energy density of the Continuum RFIC is exactly known for every α, see (2.38).On the other hand, as pointed out at the end of Section 2.4, the knowledge of the free energy density, as complete as it may be, is very far from disclosing the pathwise domain-wall structure.And understanding the domain-wall structure for the case α ̸ = 0 is, to our opinion and knowledge, an open issue.
2.6.Organization of the paper.We start, Section 3, with introducing and analyzing the simplified model, announced in the caption of Fig. 2. In particular we prove Theorem 3.3 which is the analog of Theorem 2.2 in the simplified framework.This section is definitely useful for the intuition, but it is also needed because the simplified model enters the main proof as a tool.
Our main result, Theorem 2.2, is proven in Section 4. This is a short section, because it relies on two results, Lemma 4.1 and Proposition 2.5.
Lemma 4.1 and Proposition 2.5 are proven in Section 6, after the detailed analysis of the one-sided processes l • and r • that can be found in Section 5.
In App.A we have collected several properties on Brownian motion: some are general and known, but some are less standard and more specific to our problem.In App.B we give characterizations of the sequence of Γ-extrema for the bilateral Brownian motion and more details about how this sequence is built.In App.C we show that the Continuum RFIC can be recovered as a scaling limit of the standard (i.e., discrete) RFIC.We also sketch the proof of the overlap formula (2.44).

The simplified model
We now define the simplified versions of processes l (a) and r (b) (informally mentioned in the caption of Fig. 2), which we denote by l (a) and r (b) .In this model the steps of our proofs are much less technical, but they correspond in a precise way to the steps of the proof for the Continuum RFIC.In particular Lemma 2.4 and Proposition 2.5 (except for (2.34): this is related to Remark 3.1) have strict analogs in this simplified context and one can estimate in a simple way the analog of the expression for D Γ in (2.32).
3.1.Definition and analysis of the simplified model.Recall that U Γ (x) = ε(e x +e −x ) is the potential confining processes l (a) and r (b) , see Fig. 2. We interpret it as a potential well with soft barriers at Γ and −Γ, as is attested by the shape of the invariant measure p Γ .The present heuristic consists in replacing this pair of soft barriers by a pair of hard walls placed precisely at Γ and −Γ.In other words, we rewrite equation (2.25) and the analog for the r process with a potential that is zero in the interval [−Γ, Γ] and infinite outside of it.Let us give the formal definition of l (a) and r (b) .Instead of starting l (a) at value ∞, we start it at Γ, since the infinite potential outside the pair of walls is supposed to bring l (a) instantly to Γ: we set l where L (−) t and L (+) t , for t ≥ a, are continuous non-decreasing functions that vanish at a and satisfy We refer to formula (1.14) in [30] for a representation of l (a) t in terms of (B t ).The reader who is not familiar with Skorokhod reflection problems should not get frightened at this point.In fact, the process l The simplified model has the virtue of providing explicit expressions, which are related to the structure of the Γ−extrema of B • in a very direct way: we are going to explore this next.
We recall that we have introduced in Section 2.2 the spin trajectory s (F,R)

•
-i.e., Fisher infinite disorder RG fixed point over the whole of R -via a global procedure, see in particular (2.31) (this is detailed in App.B.1).As hinted to just before Proposition 2.5, s (F,R) • may be introduced also via a local procedure, see App.B.2, in particular (B.4).For what we are doing now, it suffices to focus on the local approach.
For conciseness, we derive the explicit expression of l (a) 0 , when a is large and negative.This is the objective of the next lemma.Recall from Section 2.2 the definitions of t ↑ (Γ), t ↓ (Γ), u ↑ (Γ), u ↓ (Γ), u 1 (Γ) and a 1 .We consider the analoguous quantities for the Brownian motion B rv and set: ) as well as v 1 (Γ) := −u B rv ,1 (Γ) and a ′ 1 := a B rv ,1 .
Lemma 3.2.For every a ≤ max(s ↑ (Γ), s ↓ (Γ)), l (a) 0 does not depend on the value of a, so we write l 0 for l (a)  0 for a ≤ max(s ↑ (Γ), s ↓ (Γ)), and we have Respectively, for every b ≥ min(t ↑ (Γ), t ↓ (Γ)), r (b) 0 =: r 0 does not depend on b and As a consequence, when a ≤ max(s ↑ (Γ), s ↓ (Γ)) and b ≥ min(t ↑ (Γ), t ↓ (Γ)) we have and s (F,R) 0 = sign( m 0 ) with the convention that sign(0) = 0.These results extend to every t ∈ R by replacing B • by B (t) • everywhere, thus defining processes t → l t , t → r t and t → m t on the whole of R. In particular we have This lemma should be considered as the corresponding version of Lemma 2.4 for the simplified model, with the further advantage that, here, we are able to derive explicit expressions for the limiting processes and these expressions relate to the sequence of Γextrema.We have in particular that for a → −∞ and b → ∞ but in fact, from Lemma 3.2 one can extract substantially stronger statements.For example that, for almost every B • and for every sufficiently large ℓ one can find ℓ 0 (not depending on ℓ) and ℓ 1 (ℓ), with lim ℓ→∞ ℓ 1 (ℓ)/ℓ = 1, such that m (0,ℓ) t = m t for every t ∈ [ℓ 0 , ℓ 1 (ℓ)].
Proof.Let us assume for example that s ↓ (Γ) > s ↑ (Γ) and let us take a such that a ≤ s ↓ (Γ).Recall that v ↓ (Γ) is the (almost surely) unique time v ∈ [s ↓ (Γ), 0] such that (3.9) We proceed in three steps (see Figure 3): (1) At time s ↓ (Γ), we simply remark that l (a) is an upward record time after time s ↓ (Γ) and that and exhibits no drop of size Γ, hence l (a) evolves according to 2B • and we obtain that which is the announced value of l 0 in the case when s ↓ (Γ) > s ↑ (Γ).
Let us make the argument more explicit, by taking an analytic approach.At time s ↓ (Γ), l is in [−Γ, Γ] by definition.From this time on, l evolves as 2B • until it hits −Γ or Γ.By definition of s ↓ (Γ) and v ↓ (Γ), we have that we get that τ is not larger than v ↓ (Γ) and that l (a) τ = l (a) s ↓ (Γ) + 2(B τ − B s ↓ (Γ) ) = Γ.Now, from time τ on, l (a) evolves as 2B • reflected at Γ, until it touches −Γ.Explicitly: Since we assumed s ↓ (Γ) > s ↑ (Γ), there is no drop of size Γ in B • between s ↓ (Γ) and 0, a fortiori between τ and 0. Therefore τ ′ is not smaller than 0, we get Replacing B • by −B • , we get the corresponding result in the case when s ↑ (Γ) > s ↓ (Γ) because, as we pointed out, the result does not depend on the initial condition.The result for process r (b) follows by changing B • into −B rv • .Finally, the identity s (F,R) 0 = sign( m 0 ) follows from the analysis in Appendix B, specifically it is just a more concise version of (B.4).□

3.2.
Main theorem for the simplified model.Now we are going to state and prove the analog of our main theorem (Theorem 2.2) for the simplified model.In fact, due to Remark 3.1 we formulate only the analog of (2.33) (the analog of (2.32) follows), but the result is more precise in terms of the large Γ dependence: Proof.It is not difficult to show that the ergodic result (2.33) holds for the simplified model, in the sense that We only sketch the proof of (3.15) (because it is very close to the argument in the proof of (2.33) itself, see Section 6.2).The idea is to show first that almost surely by an adequate control of the domain where the two integrands differ: see (3.8) (and the observation right after it), moreover the substitution of s (F ) t with s (F,R) t is also straightforward because they can possibly differ only when t ∈ [0, u 1 (Γ)] a.s.. Finally, one goes from (3.16) to (3.15) by applying (the time-continuous version of) Birkhoff's Ergodic Theorem.
To conclude the proof of Theorem 3.3, we analyze the expectation in the right-hand side of (3.15).Observe that l 0 is measurable with respect to F((B t ) t≤0 ), while r 0 is measurable with respect to F((B t ) t≥0 ), hence they are independent.Furthermore, they follow the uniform law on [−Γ, Γ] (see Lemma A.2(ii)).Now the key point is that the simplified model captures exactly Fisher's strategy -or the other way around -in the sense that s (F,R) 0 = sign( m 0 ), as stated in Lemma 3.2.Therefore: where Li 2 (•) is the Polylogarithm of order 2 that has an analytic behavior at the origin, with Li 2 (x) ∼ x for x near 0 [38, (25.12.10)].□

Proof of Theorem 2.2
We give the proof of Theorem 2.2 assuming the validity of Lemma 2.4 and Proposition 2.5, stated in the introduction, and of Lemma 4.1 which is stated below.The first result is proven in the next section, see Proposition 5.8, and to the other two we devote Section 6.A look at Proposition 2.5 makes clear that Theorem 2.2 follows if we obtain an adequate upper bound on the expression for D Γ given in (2.32).As already discussed in Section 2.3 (and proven in Proposition 5.8), the law of (l 0 , r 0 ) is simply p Γ ⊗ p Γ , with p Γ defined in (2.26), but knowing the law of (l 0 , r 0 ) is not of much help because we need to relate (l 0 , r 0 ) and s (F,R) 0 in a pathwise sense, that is for (almost) every trajectory of B • = (B t ) t∈R .In our study of the simplified model, we established that s (F,R) 0 is equal to the sign of m 0 = l 0 + r 0 .We are going to show that this remains essentially true for the real model.This comes from the next lemma and it is made completely explicit in the corollary that follows it.Lemma 4.1.For every κ > 0, there exists a constant C κ such that, with probability 1 − O(Γ −κ ), we have: We postpone the proof of Lemma 4.1 to Section 6.1.Of course in Lemma 4.1 time 0 can be replaced with any time t ∈ R: this is just a matter of using B (t) • instead of B • and the event underlying probability 1 − O(Γ −κ ) changes accordingly.
Moreover, using that l 0 and r 0 are uniformly distributed over [−Γ, Γ], from Lemma 4.1 for κ = 1 we directly obtain: Now, we are ready to finish the proof of our main theorem.
Proof of Theorem 2.2.We estimate the expression for D Γ in (2.32).We recall that with the convention sign(0) = 0 and we introduce: We handle the expression for D Γ in (2.32) when s (F,R) 0 is replaced by s (m) 0 there, so using the independence of l 0 and r 0 : where in the last step we used that for every η > 0, every Γ sufficiently large and every , with u Γ the density of the uniform probability on [−Γ, Γ], and To conclude we use the fact that (1 + exp(•)) −1 is bounded and Corollary 4.2, which yield The proof of Theorem 2.2 is therefore complete.□ over all stochastic processes (S t ) t∈[0,ℓ] defined in the same probability space as B • and taking values in {−1, 0, +1}.By the same argument of proof, also in this case m (0,ℓ) • can be replaced in the limit by m • and the analog of Theorem 2.2 holds with D Γ now equal to (4.7) Remark 4.4.One can of course make the estimate (4.4) precise to all orders: here we give the second order in the expansion.The law of l 0 + r 0 , i.e. p Γ * p Γ , is which is an even distribution, close to the convolution of the uniform law over so that, recalling (3.17) for the contribution due to the approximation with uniform laws, we obtain that the expression in (4.4) is equal to That is

Analysis of the one sided processes
Recall the definitions of Section 2.3, notably that a < b, L (a) t is defined in (2.20) and that R (b)  t is defined in (2.21).Also R t := R (b) −t and B rv t := B −t are introduced in Section 2.3.The notion of strong solution to a stochastic differential equation with time running from 0 to ∞ is given for example in [40, p. 366] and it directly generalizes to the case in which t ∈ [s, ∞), for every s ∈ R. We say that (L t ) t>a is a strong solution of (2.22) if (L t ) t≥a ′ is a strong solution of (2.22) for every a ′ > a.In particular, (L t ) t≥a ′ is adapted to the filtration (σ t ) t>a is a strong solution of (2.22) and satisfies lim t↘a L (a) t = ∞ a.s.. (R t ) t>−b is a strong solution of (2.23) and satisfies lim t↘−b R t = ∞ a.s.. Proof.The proof is given in two steps: the first one exploits a time inversion trick on the Poisson process (η t ), via the equilibrium version of this process, to reduce the analysis of R (b) to the analysis of L (a) .Then the analysis of L (a) is performed by exploiting that L (a) (see see (2.20)) is a ratio of two stochastic processes defined on the same probability space and adapted to the same Brownian filtration.It is then practical to consider these two stochastic processes as one stochastic process taking values in R 2 .We then show that, by Itô formula, this two dimensional stochastic process solves a linear stochastic differential system: this system is treated in detail in [11,Sec. 2], where it is shown that both components are strictly positive except at the initial time and that the ratios of the two components solve differential equations.In fact, the two ratios solve the same stochastic differential equation, except for the sign of the driving Brownian motion and for the initial condition.The initial condition of the ratio we consider is ∞, which makes a bit unpleasant the analysis, but this disappears if one considers the other ratio that has initial condition 0 and directly falls into the standard theory of solutions to stochastic differential equations.
The fact that solutions are strong is no surprise: as we just pointed out, the stochastic differential equations we deal with are ultimately even coming from a linear system.But we stress it because it implies a number of useful properties: for example that, given s ∈ (a, b), (L (a)  t ) t∈(a,s] and (R (b) t ) t∈[s,b) are independent.Let us therefore begin by arguing that it suffices to establish the result for L (a) • .As we have anticipated, for this it is practical to work with the equilibrium process (η t ) t∈R : this is defined by stipulating that the wall locations are given by an homogeneous Poisson process on R with intensity ε = exp(−Γ) and the sign of the spins between walls is determined once the sign η 0 of the domain that contains the origin is chosen.The sign of η 0 is a Bernoulli of parameter 1/2, independent of the Poisson process.We call P eq Γ the law of this process and we remark that (5.1) In fact, since P eq Γ (η a = 0) = P eq Γ (η a = 1) = 1/2, the process (η s ) a≤s≤t under P eq Γ (• | η a = 0) has the same law as under P Γ , so (5.1) holds.Analogously Now we remark also that (η −s ) s∈R and (η s ) s∈R have the same finite dimensional laws, and if we set η −s := lim s ′ ↘−s η −s ′ then (η −s ) s∈R and (η s ) s∈R have the same law, so and let us remark that the symmetry is more evident in the case a = −b.These expressions show that it suffices to prove the statement for L (a) • and this is what we do next.For the rest of the proof we can use the more compact expression (2.20) for L (a)  t and let us note that lim t↘a L (a)  t = ∞ a.s. is immediate from (2.20) because s t = (−1) N t−a , hence P Γ (η a = 0) = 0. We set for j = 1 and 2 (5.4) The process (X 1 (t), X 2 (t)) t≥a is clearly adapted to the filtration (σ(B s −B a , a ≤ s ≤ t)) t≥a and we claim that it solves the linear SDE system dX 1 (t) = εX 2 (t) dt and dX 2 (t) = (εX for t ≥ a with (X 1 (a), X 2 (a)) = (1, 0).Since L t = X 1 (t)/X 2 (t), the statement follows by applying Itô formula (note that we have partly kept the same notations as in [11]: in our case σ = 2 and L t = 1/Y t , σ and Y • defined in [11]).In order to establish (5.5) we remark that η t = 1 ηt=1 , 1 − η t = 1 ηt=0 and we set X j (t) := e εt exp −2 t a η s dB s 1 ηt=j−1 . (5.6) By rewriting in integral formulation we have for t > a and with X(t) := X 1 (t) + X 2 (t) where X(s-) = lim t↗s X(t).Moreover from the first to the second line the Brownian integral term and the Itô term disappear because the integrands contain η s (1 − η s ) = 0 and in the next step we have used , where N ε,• is the martingale associated to the Poisson process N • , i.e.N t = εt + N ε,t : in particular, M t ∈ L p for every p).Since X(s)(1 − 2η s ) = X 1 (s) − X 2 (s), by taking the expectation with respect to the Poisson process (recall that X j (t) = E Γ [X j (t)]) we readily see that which is the first equation in (5.5).For the second one we write X(s-) dη s , (5.9) where η s disappears in Brownian integral and in the Itô term because η 2 s = η s .By using again X(s)(1 − 2η s ) = X 1 (s) − X 2 (s) and by taking the expectation we recover the second equation in (5.5).This completes the proof of Lemma 5.1.□ Remark 5.2.Lemma 5.1 is easily generalized to arbitrary fixed boundary conditions: if instead of boundary conditions +1 on the left we have −1 then L (a) t solves the very same SDE, but L (a) a = 0 (no need to taking the limit in this case).Exactly the same statement holds if we change the boundary on the right.And it is useless to repeat the arguments we have just detailed in the case of plus boundary conditions.In fact it suffices to remark that if we use that (η t ) t∈R has, under P eq Γ , the same law as (1 − η t ) t∈R , so (5.1) becomes (5.10) t becomes the expression we need with −1 left boundary condition, except that B • is replaced by −B • .But, as already pointed out in the introduction, Itô formula yields that 1/L (a)  t solves the same SDE as L (a) t with B • replaced by −B • , and with the obvious change in the initial condition.Therefore we see that the change in the boundary condition for the spin system just leads to changing the boundary condition for the SDE.
For what follows we step to work with l (a) t = log L (a) t and r (b) t = log R (b) t , introduced in (2.24).
Remark 5.3.Let us set a = 0 for conciseness.One can integrate the first equation in (5.5) to obtain which relates the (almost sure) exponential rate of growth of X 1 (•) (that is the free energy density up to the additive constant ε, see (5.4)) with the ergodic properties of l (0) • .An analogous argument can be applied to X 2 (t) and this leads to a formula that differs from (5.11): nonetheless, one can show that X 1 (•) and X 2 (•) have the same rate of growth (see [11,Th. 1.1]).
Without loss of generality, we just consider l (a)  t : one readily checks that for every t ≥ a.It is the unique strong solution of this SDE, but since l (a) a = ∞ we need to make this statement precise.We are therefore going to look at the problem with more general initial condition and establish some monotonicity properties.This will allow us also to deal with a → −∞ and with the construction of a (unique) solution on the whole of R. Some of the technical results that we state and prove here will be of use also outside of this section.
We start with a comparison lemma.
Lemma 5.4.Let (ω t ) t≥0 be a (F t ) t≥0 -Brownian motion and f * : R → R be a locally Lipschitz function.Assume x t , x * t , a t , f t to be (F t ) t≥0 -adapted real-valued continuous processes such that almost surely, for every t ≥ 0, We put the constant 2 before ω t only for notation reasons in the applications of this lemma.When a s (resp.f s ) is a deterministic function of s (resp. of (s, x s )), this lemma is a particular case of the general comparison theorems for one-dimensional diffusions, see [28].
Proof.By considering the processes x t and x * t stopped at inf{t ≥ 0 : |x t |+|x * t | ≥ n} for n = 1, 2, . .., we can assume that they are bounded.Then there exists K > 0 such that for every Then, considered as a (F t ) t≥0 -continuous semimartingale the process x t −x * t has vanishing local times.Applying the Tanaka formula ( [40], Theorem VI.1.2),we have max(0, We conclude by Gronwall's Lemma that max(0, Let us introduce the unique strong solution l (a,x) t of (5.12) with initial condition x ∈ R, that is the only continuous process, adapted to the filtration σ(B s − B a , a ≤ s ≤ t), t ≥ a, that satisfies a.s.
for every t ≥ a.
We collect in the next lemma useful upper and lower bounds on l (a,x) t that follow by applying Lemma 5.4.Lemma 5.5.For every x ∈ R and every t > a we have l (a,x) with Moreover for every t > a a.s.we have The idea behind the proof of this lemma is that we are dealing with a diffusion between two barriers.In particular we can bound the solution from below in terms of the solution found when we suppress the left barrier (and this bound will not be bad as long as the solution without left barrier does not approach the region where the left barrier was located).And when there is only one barrier the solution is explicit.
Proof.For the lower bound, we argue that l (a,x) s was defined such as to satisfy the stochastic differential equation Itô formula indeed yields that this SDE is equivalent to the SDE d exp −l (a,x) t + 2B t = ε exp(2B t ) dt, which can be integrated immediately.By applying Lemma 5.4 we establish the lower bound.For (5.20) it suffices to observe that and the passage to x → ∞ is straightforward.
For the upper bound we first exploit the lower bound that we have just established to obtain that for t > a l (a,x) hence, by Lemma 5.4, we have that l (a,x) ) t≥a as the unique solution of for every t > a: in the second line we have used that which can be verified by plugging in the explicit expression for l (a,x) u given in (5.17) (θ x t is defined in (5.19)).The proof of the upper bound is now completed by remarking that the stochastic process explicitly given in (5.18) solves (5.25): in fact (5.25) is the same as (5.22) with x + 2(B t − B a ) replaced by θ x t and can be solved in the same way.We are left with proving (5.21).For this we remark that from (5.18) and (5.19) we have e 2(Bs−Ba) ds .
(5.27) From (5. 19) we readily see that lim x→∞ (θ a e 2(Br−Bs) dr ds a.s. and (5.21) follows by rearranging the terms.□ We give here also another result in the same spirit as Lemma 5.4 that is useful in order to extend the bounds in Lemma 5.5 to random times that are not stopping times.In fact, the result is fully deterministic: Lemma 5.6.Given T ∈ R and a continuous function t → b t we consider a continuous function ℓ • that solves for every t ≥ T , for a given value ℓ T .Then for every t ≥ T , Proof.Note that we may apply (5.29) to −ℓ and −b instead of ℓ and b and obtain (5.30).
Then it is enough to show (5.29).Call u t the right-hand side in (5.29) and set x t := u t − ℓ t .One can directly check by taking the time derivative of u t − 2(b t − b T ), i.e. the logarithmic part of the expression, and by integrating that (5.31) Since x T = 0, we get (5.29) and complete the proof.□ Proposition 5.7.For every x < x ′ we have l (a,x) t < l (a,x ′ ) t for every t ≥ a, hence l (a,−∞) t := lim x↘−∞ l (a,x) t and l (a,∞) t := lim x↗+∞ l (a,x) t are well defined for every t.Moreover, almost surely we have that t → |l (a,±∞) t | are bounded over any compact subset of (a, ∞), lim t↘a l (a,±∞) t = ±∞ and for every s, t with t > s > a, we have Proof of Lemma 4.1.We recall that the stationary process (l t ) t∈R is defined via Proposition 5.8 and that the law of l t is p Γ .Recall also from (3.4) the definition of s ↑ (Γ) and s ↓ (Γ) which are stopping times for the reversed Brownian filtration.Recall from (3.9) that v ↓ (Γ) is the unique time in [s ↓ (Γ), 0] such that Pick κ > 0. In strict analogy with the proof of Lemma 3.2, our program is resumed in three steps: on the event {s ↓ (Γ) > s ↑ (Γ)} and excluding an event of probability O(Γ −κ ) • Step 2 : exploiting that there is a rise in B • of size Γ between times s ↓ (Γ) and v ↓ (Γ) to show that Actually, in the proofs of Step 1, of Step 2 and of the lower bound in Step 3 we will work without assuming that s ↓ (Γ) > s ↑ (Γ): the statements remain of course true on this event, which has probability 1/2.

Proof of
Step 2: upper bound.The idea is to use a bound on the modulus of continuity of B • : we first give a sketch of the argument.Choose F > 0, a > 0 and λ > 0 such that 2λ ≤ ae F .Assume that B • cannot make a rise larger than λ inside any interval of length a.When l • is above Γ + F , then the repulsion, which acts essentially like −e F dt, makes l smaller by at least ae F in an interval of length a, while B • can make it larger of at most 2λ.Thus l • cannot stay above Γ + F over an interval of length longer than a, so it cannot grow beyond Γ + F + 2λ.
In order to fill in the details, we start with: Lemma 6.1.Let t > 0 and b : [0, t] → R be a continuous function.Consider ℓ : [0, t] → R solution of the ODE for every s ∈ [0, t] (and for a given initial condition ℓ 0 ).Let F > 0, a > 0 and λ > 0.
Assume moreover that (1) (2) λ ≤ εa sinh(Γ + F ).Then, provided that the initial condition satisfies ℓ 0 ≤ Γ + F , we have Now, we argue that s ≤ ϱ + a.In fact, if s > ϱ + a then, by applying (6.13) to u = ϱ + a and by assumption (2), we obtain that which is impossible by the definition of ϱ and because s > ϱ+a.We conclude that s ≤ ϱ+a and therefore inequality (6.13) applies to u = s, hence This concludes the proof of Lemma 6.
where in the second line we used that B 0 = 0. Note that where C κ,3 = log e C κ,2 + C κ,6 .
Proof of Proposition 2.5.First we observe that we can replace s (F ) with s (F,R) .In fact these two trajectories, as discussed at length in App.B, coincide on (u (6.25) From Lemma 5.10 we derive the following Lemma, which quantifies the convergence in (6.25).Lemma 6.3.Choose any Γ > 0, any trajectory B • and any real numbers a < t < b.Then where we denoted A ε (t) = 2 arctanh e −2εt for every t > 0.
We observe that A ε is integrable on (0, ∞) and Proof.Using (6.24) and (6.25) we have where for the first inequality we have used that (1 + exp(•)) −1 is 1 2 -Lipshitz and for the third inequality we have used Lemma 5.10.□ Now, we are ready to prove (2.32) (with, as argued, s (F,R) instead of s (F ) ).From Lemma 6.3 we have In view of (6.28) the proof of (2.32) is straightforward because, by time invariance of the Brownian motion, By (6.28) we have also that (2.33) follows if we show that almost surely This is a direct application of the (continuous-time) Birkhoff Ergodic Theorem (see for example [29, p. 10]).In fact, observe that the family of operators (φ t ) t∈R defined on the Wiener space, i.e. the space of continuous functions endowed with the topology of uniform convergence over compact sets, by φ t (B • ) = B (t) • where we recall that is an ergodic flow with respect to the (bilateral) Wiener measure P. If we set we have that f (•) ∈ L ∞ (P).Therefore Birkhoff Ergodic Theorem yields that a.s.
To prove the convergence in µ Γ,B•,ℓ -probability of variable I ℓ (s t ) t∈[0,ℓ] , see (6.23), we are going to control its variance under µ Γ,B•,ℓ .Then we start by computing the second moment.Using the fact that for 0 ≤ t 1 < t 2 ≤ ℓ and σ 1 , σ 2 ∈ {+1, −1} we have we establish that By using Lemma 6.3 we bound the second moment in (6.34) by i.e., , plus an error term which can be bounded by where we used that if x, x ′ , y, y Then we observe that, by (6.28), we have also that Therefore, we have shown that the variance of I ℓ (s t ) t∈[0,ℓ] under µ Γ,B•,ℓ is O(1/ℓ).In particular it converges to 0 as ℓ goes to infinity and the proof of Proposition 2.5 is complete.□ Appendix A. Some results about Brownian motion First of all, we need a control on the continuity modulus of Brownian motion: there exists a constant c 1 > 0 such that for every t > a > 0 and λ > 0, we have where 3 may be replaced by any number larger than 2 (for a proof, see [14]).The rest of the results of this appendix are more specific and related to the sequence of Γ-extrema.Recall that, by Brownian scaling, (t ↓ (Γ), u ↓ (Γ)) for every λ ≥ 0. In particular, E[e λt ↓ (Γ) ] < ∞ if λ < π 2 8Γ 2 .(ii) Let (u n (Γ)) be the sequence of the times of Γ-extrema of (B t ) t≥0 .Then (u n+1 (Γ) − u n (Γ)) n=1,2,... is an i.i.d.sequence, with Laplace transform (λ > 0) where J 0 (•) denotes the Bessel function of the first kind [38, (10.2.2)] with index 0 and j 0 > 0 is its smallest positive zero.
We note that the first upper bound in (A.4) holds uniformly in Γ, and the second one involves Γ and will be useful when Γ is small.Proof.Only (iii) needs a proof because (i) and (ii) are proven in [36].For the first part of (iii) we observe that, by Lévy's identity, β t := sup s≤t B s − B t , t ≥ 0, is distributed as a reflected Brownian motion.Then t ↓ (Γ) becomes the first hitting time of Γ by β and u ↓ (Γ) To complete the proof of (ii), we observe that ( and the proof of (ii) is complete.
Proof of (iii).At first by (A.4), the proof of (A.9) is reduced to show that for all small c 2 > 0, sup Since {(r, e ′ r ) : e ′ r ̸ = ∂} forms a Poisson point process with intensity measure dr ⊗ n(dγ, max γ < Γ)) and independent of H, we deduce from the exponential formula that where in the last equality we use the Williams description of the Itô measure (see [40], Theorem XII.4.5) and R 3 denotes as before a three-dimensional Bessel process.By (A.4), for 0 < c 2 < min(j Proof of (iv).The proof of (A.10) bears some similarities with that of (A.20).Like in the proof of (iii), we have for a suitable choice of the positive constant c 9 independent of Γ and h.In the second inequality we have used the existence of c 8 > 0 such that λ 0 min(1, m 2 ) e −2(λ−m) dm/m 2 ≤ c 8 min(1, λ −2 ) for every λ > 0. Going back to (A.27), we see that the expectation term in the left-hand-side is bounded above by c 9 .This implies (iv) and completes the proof of Lemma A.2. □  (C.4) where with respect to (C.3) there is the minor change of summing from 0 up to N −1 instead of from 1 to N = ℓ/∆ and there is, above all, the introduction of the probability measure E Γ,∆ under which (σ j ) is a Markov chain with two states (−1 and +1) and transition probabilities Q(+1, +1) = Q(−1, −1) = 1/(1 + ϵ) and Q(+1, −1) = Q(−1, 1) = ϵ/(1 + ϵ), ϵ = exp(−2J) = ∆ exp(−Γ).This is a consequence of the fact that Z N,J,0 is equal to (1, 0)Q N (1, 1) t times (1 + ϵ) N .We recall that σ 0 = 1 and we remark that, in matrix notation, (1, 0) t corresponds to spin up and (0, 1) t to spin down.
Before going into the convergence issues let us remark that it suffices to prove (C.4) with exp(•) replaced by exp(•) ∧ L, for every L > 0. This is because if we call Z ∆ the partition function in the left-hand side of (C.4) and Z the one in the right hand side, we have that Z ∆ = E Γ,∆ [exp(H)] with a suitable choice of H. Hence (C.8) where we have used that the hypotheses on ω 1 yield the existence of c > 0 and t 0 > 0 such that E[exp(tω 1 )] ≤ exp(ct 2 ) for |t| ≤ t 0 .In the same way we obtain that EE Γ [exp(2H)] is bounded by exp(2(α + 4)ℓ).This concludes the argument that shows that it suffices to show (C.4) for exp(•) replaced by exp(•) ∧ L.
Proof.We just give a sketch of the argument.First of all we remark that, by scaling properties, if we replace

2. 3 .
A two sided view of the model.It is helpful from the technical viewpoint to consider the model on [a, b], any a < b, instead of simply [0, ℓ].In view of this generalized setup it is natural to introduce the standard bilateral Brownian motion B = (B t ) t∈R .For the general case of the model on [a, b] the free process is (s t ) t≥a with s t = (−1) N t−a .The partition function of the Continuum RFIC on [a, b] becomes Z Γ,B•,a,b := E Γ exp b a s t dB t ; s b = +1 , (2.15) and we note that we set α = 0.The stochastic integral in (2.15) should be understood as b−a 0 s a+t dB (a) t , with B (a) t := B t+a − B a , t ≥ 0. (2.16)The associated Gibbs measure is denoted by µ a,b = µ Γ,B•,a,b .
) and Z Γ,B•,a,b e Ba−B b is an equivalent partition function since e Ba−B b does not involve the spin configuration, hence it yields the same Gibbs measure µ Γ,B•,a,b .By the Markov property of the Poisson process we have for every t ∈ (a, b)

Remark 2 . 3 .
This symmetry becomes more transparent if one considers the case b = −a = ℓ and if one thinks of the statistical mechanics origin of the model.Moreover (2.22) and (2.23) imply that (L (−ℓ) t) t≥−ℓ and (R(ℓ)

Figure 2 .
Figure 2. The plot of the potential U Γ on the left and of the invariant density p Γ on the right, both for Γ = 20.The strong proximity of U Γ with ∞1 [−Γ,Γ] ∁ and of p Γ with the uniform measure on [−Γ, Γ] motivates the introduction of the simplified model of Section 3 that is going to be an important tool in our analysis.
Then, the process l (a) is constrained to remain in [−Γ, Γ] ever after and l(a) t , L (−) t , L (+) t t≥ais the unique solution of the following Skorokhod problem (see[30, equation (1.6)]) as follows: l (a) is driven by the Brownian motion B (times 2) except if it saturates at Γ or at −Γ, in which case it is forbidden to cross these boundaries.The process r (b) • behaves similarly but in reversed time: the process ( r (b) −t ) t≥−b is started at r (b) b = Γ and is given by the solution of the Skorokhod problem (3.1), except that B is replaced by −B rv there, with the same constraint to stay in [−Γ, Γ].We also define m (a,b) a ≤ t ≤ b.These two simplified processes, and the associated m process, constitute what we call the simplified model.Remark 3.1.We do not know whether there exists a spin process analogous to process (s t ) t∈[a,b] under law µ Γ,B•,a,b for which the probability that the spin at t is equal to +1 would be equal to (1 + exp(− m (a,b) t )) −1 (recall (2.27)).

Lemma C. 2 .
For α = Then a.s.x t ≤ x * t for every t ≥ 0. Similarly if x 0 ≥ x * 0 and f t ≥ a t + f * (x t ), then a.s.we have x t ≥ x * t for every t ≥ 0.