Multi-level pinning problems for random walks and self-avoiding lattice paths

We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $\sum_{j\geq 0}\epsilon_j N_j$, where $N_j$ is the number of visits to the state $j$ and $\{\epsilon_j\}$ is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance $\sigma^2$ of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence $\{\epsilon_j\}$, localization (respectively delocalization) occurs if $\sigma^{-2}\sum_{ j\geq0}(j+1)\epsilon_j\geq\delta^{-1}$ (respectively $\le \delta$), for some universal $\delta<1$. Our finding is reminiscent of the classical Bargmann-Jost-Pais criteria for the absence of bound states for the radial Schr\"odinger equation. The core of the proof is a recursive argument to bound the free energy of the model. Our approach is rather robust, which allows us to obtain similar results in the case where the random walk trajectory is replaced by a self-avoiding path $\gamma$ in $\mathbb Z^2$ with weight $\exp(-\beta |\gamma|)$, $|\gamma|$ being the length of the path and $\beta>0$ a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models.


Introduction and motivations
Consider a one-dimensional integer-valued symmetric random walk starting at zero, conditioned to stay non-negative.If the walk has a reward ε > 0 for each return to zero, it is a classical fact that there exists a critical value ε c such that for ε > ε c the random walk has a positive density of returns to the origin while for ε < ε c entropic repulsion prevails and the density of returns is zero; see e.g.[12] and references therein.This is often called a wetting transition [11].The critical parameter ε c depends crucially on the single-step variance σ 2 ; in simple examples such as the symmetric walk with increments in {−1, 0, +1} one finds that ε c scales linearly in σ 2 as σ 2 → 0 [14].
In this work we consider a natural generalization where the pinning at the origin is replaced by a long range pinning potential ε = {ε j } j 0 , where ε j 0 is the reward for a visit to the state j 0. To be specific, for a trajectory γ of length L, define where N j (γ) is the number of visits to state j.Define also the free energy where E + 0,L (•) stands for the expectation w.r.t. to the path measure conditioned to γ 0 and γ 0 = γ L = 0.The existence of the limit follows by sub-additivity.With this notation the localized (resp.delocalized) phase is characterized by f (ε) > 0 (resp.f (ε) = 0).Under mild assumptions on the random walk kernel and on the pinning sequence ε, we prove that the wetting transition occurs at a critical value ρ c of the ratio (j + 1)ε j and that ρ c ∈ (a, b) for universal constants a, b > 0.
As far as we know, this is the first analysis of the wetting transition for a multi-level pinning problem of the general form (1.1).We refer to [5,16] for previous studies of certain specific models of random walks with pinning on several layers.It is interesting to note the analogy between our condition for delocalization and the classical Bargmann-Jost-Pais [1,15,17] criteria for the absence of bound states for the radial Schroedinger equation; cf.Remark 2.4 below for a discussion of this point.
The most challenging part of the proof is to show delocalization for ρ small.That requires establishing an upper bound on the partition function where the sum runs over non-negative trajectories returning to the origin at time L and w(γ) is the probability of γ.Using the strategy outlined below we prove that where C is a universal constant and Z 0,L = γ w(γ), the sum being over all trajectories returning to the origin at time L. Clearly, E + 0,L e Φ = Z Φ,+ 0,L Z 0,L P 0,L (γ 0) −1   where P 0,L (γ 0) is the probability that a path returning to the origin after L steps remains non-negative.Well known bounds show that P 0,L (γ 0) −1 = O(L), so that the estimate (1.2) establishes delocalization.
To prove (1.2) we argue as follows.The first step is to decouple the problem into a collection of independent pinning problems, one for each height level j = 0, 1, . . . .More precisely, let ρ j := (ρσ 2 ) −1 (j + 1)ε j so that j 0 ρ j = 1.Then, Jensen's inequality implies that where Z +,κ j 0,L = γ 0 w(γ)e ρσ 2 N j (γ)/(j+1) is the partition function of a random walk returning to the origin after L steps, pinned at height j with pinning strength κ j = ρσ 2 /(j + 1).The next step is to show that, if the parameter ρ is small enough, then uniformly in the law of the random walk and in the height j, one has Z +,κ j 0,L CZ 0,L .Using j ρ j = 1, this bound implies (1.2).The main idea of the proof goes as follows.With a natural inductive argument we show that, for all L larger than a critical "diffusive" scale L c (j) ∼ (j + 1) 2 /σ 2 , one has Z +,κ j 0,L CZ 0,L , provided that the same holds for L = L c (j).The base case of the induction is solved by a fine analysis based on careful local limit theorem estimates.It is only at this stage, that is when L = L c (j), that we need to take the parameter ρ small enough.
The method outlined above is rather accurate in finding the threshold for the wetting transition.In this respect, we remark that a direct "energy vs. entropy" argument such as the one used in [2] would fail to capture the right dependence in the parameter σ 2 for instance.Moreover, our method is robust enough to admit an extension to the setting of self-avoiding paths, as we discuss below.
One of the motivations for this work stems from the mathematical analysis of contour models arising in low-temperature two-dimensional spin systems and related interface models.In this context the random walk is replaced by a self-avoiding and weakly selfinteracting random lattice path with an effective diffusion constant σ 2 ∼ e −β , where β is the inverse temperature.Here the analog of the pinning strength ε j above typically decays like e −αβ(j+1) for j → ∞ with α > 1.Whether such long range potential is able to localize the contour is a key question in the analysis of large deviations problems such as e.g. the Wulff construction for the 2D Ising model [8] and for the (2 + 1)-dimensional Solid-on-Solid model [3].We refer the interested reader to [13] for more details.
Our approach can be applied in principle to this setting.In Section 4 below we work out the details of this extension in the simplified case where the self-avoiding path has no additional self-interaction.The general case has been recently solved in [13], with stronger results, with a very different approach.The main idea of [13] consists in constructing, out of the self-interacting contour path, an effective random walk together with a renewal structure and then prove delocalization for the latter.

Models and results
2.1.Random walks and pinning.We consider a class of symmetric and irreducible random walk kernels on Z with variance σ 2 .Since we are interested in the regime of small σ 2 , we will make the assumption σ 2 ∈ (0, 1  2 ].Definition 2.1.For a fixed constant c 0 ∈ (0, 1] and σ 2 ∈ (0, 1  2 ], we call P(c 0 , σ 2 ) or simply P(σ 2 ) the set of all symmetric probabilities p : In particular, any p ∈ P(σ 2 ) satisfies p(0) 1 − σ 2 1 2 , and the associated random walk is irreducible.Below we shall restrict ourselves to random walk kernels in the class P(σ 2 ).While we do not believe this to be the largest possible class for our results to hold, the above assumptions turn out to be very convenient from the technical point of view.At the same time, they include a wide range of interesting models.Two key examples to keep in mind are: 1) the symmetric nearest neighbor walk with p(±1) = σ 2 2 , p(0) = 1 − σ 2 and p(k) = 0 otherwise, referred to as the binomial walk, and 2) the geometric walk with p(k) = 1 Z β e −β|k| , k ∈ Z, where β ∈ (0, ∞) is the unique positive solution of and We refer to this as the SOS walk at inverse temperature β, from its relation with the so-called Solid-On-Solid model.
We call P i the law on trajectories of the random walk starting at i ∈ Z.Let Ω 0,L = {γ = (γ 0 , . . ., γ L ) : denote the set of trajectories which start and end at zero, and define the partition function We write P 0,L for the law of the walk conditioned to Ω 0,L , that is for any γ ∈ Ω 0,L : For a fixed integer j 0, consider the paths Ω +,j 0,L that stay above height −j: 2) The number of contacts with level zero is given by For any ε > 0 we consider the probability measures ), and the corresponding expectations E +,j 0,L , E ε,+,j 0,L .Theorem 1.There exist constants a b > 0 and c > 0, such that the following holds for any integer j 0, any σ 2 ∈ (0, 1  2 ] and any random walk p ∈ P(σ 2 ): i) If ε aσ 2 /(j + 1), then for all L large enough E +,j 0,L e εN exp c σ 2 L (j + 1) −2 . (2.4) ii) If ε bσ 2 /(j + 1), then: It is well known that the ratio Z 0,L /Z +,j 0,L appearing in (2.5) is O(L) as L → ∞; see (3.14) below.In particular, the bounds in Theorem 1 imply that a wetting transition occurs at a critical value ε c that satisfies b ε c σ −2 (j + 1) a, with constants a, b that are independent of σ 2 and j and independent of p ∈ P(σ 2 ).This extends well known results in the case j = 0; see e.g.[6,7,12,14].
Remark 2.3.In Proposition 3.5 below we show that the upper bound (2.5) can be upgraded to the following bound independent of L, but with possibly non-optimal dependence on j, σ 2 : for some constant K = K(j, p), p ∈ P(σ 2 ), whenever ε bσ 2 /(j + 1).
Next, we consider a more general interaction with the wall.For any γ ∈ Ω +,0 0,L , define the potential where {ε j } is a given nonnegative sequence, and let Theorem 2. There exist absolute constant a, b, c > 0 such that, for any σ 2 ∈ (0, 1  2 ] and any random walk p ∈ P(σ 2 ), the following holds: i) For any integer d 0 such that we have for all L large enough.ii) If the sequence {ε j } satisfies (2.10) Notice that since j d in the summation (2.8), the condition for localization is slightly stronger than the bound ∞ j=0 (j + 1)ε j aσ 2 that would be sharp in view point (ii).However, in many natural cases of interest, condition (2.8) is actually rather sharp; see Corollary 1 below.Remark 2.4.Our criterion (2.10) for the absence of a localized phase bears some similarity with the one derived in [15] by Jost and Pais (cf.also the more general Bargmann's bounds in [1,17]) to exclude bound states for the Schrödinger equation Eψ in an attractive central potential V 0 in R 3 or, after moving to radial coordinates, for the Sturm-Liouville problem on the half-line In [15] it was proved in fact that if ∞ 0 dr r|V (r)| < 1 then there are no bound states.The connection between the pinning problem and the bound state problem goes as follows.Let γ(•) be the random walk on Z with law p ∈ P(σ 2 ) and let E 0 (•) denote the average over the trajectories of γ(•) starting at the origin.If τ denotes the hitting time of the half-line (−∞, −1] then we can write where F (x) = j 0 ε j 1(x = j).If we pretend that the random walk behaves like a Brownian motion with the correct diffusion constant and we replace F with V (x) := −F ( x ), then, using the Feynman-Kac formula, we get that the r.h.s.above has the form e −LH (0, 0)/p(0, 0; L), where H = −σ 2 d 2 dx 2 +V (x) acts on L 2 ((−1, +∞)) with Dirichlet boundary conditions and p(x, y; L) is the transition probability density for the Brownian motion killed at −1.If the equation Hf = λf has a solution bounded in L 2 (of course with a negative eigenvalue λ) then the above ratio should diverge exponentially fast in L. The Jost-Pais criterium says that this cannot be the case if +∞ 0 dr r|V (r)| < σ 2 , which is indeed analogous to our condition j 0 ε j (j + 1) < bσ 2 .Notice that the absence of bound states does not guarantee that the ratio (2.12) stays bounded in L. In this sense our result is stronger.
The above theorems allow us to identify rather precisely the critical point of the wetting transition for the generalized pinning problem described by (2.7) when the sequence ε j is given.For the sake of definiteness we mention only two types of sequences below, the power law and the exponential law.It is immediate to deduce the following corollary.
Corollary 1.Let the sequence ε 0 j be either the power law ε 0 j = (j + 1) −2−δ or the exponential law ε 0 j = e −δj , for some δ > 0. Then there exist constants a > b > 0 and c > 0 such that the following holds for any σ 2 ∈ (0, 1  2 ] and any p ∈ P(σ 2 ): i) If the pinning sequence is ε j := a σ 2 ε 0 j , then the walk is localized: for L large enough, ii) If the pinning sequence is ε j := b σ 2 ε 0 j , then the walk is delocalized: On the other hand, suppose that ε 0 j = (j + 1) −2+δ , for some δ > 0.Then, for any b > 0, for any σ 2 > 0, any walk p ∈ P(σ 2 ) is localized by the pinning sequence ε j = b σ 2 ε 0 j .2.2.Self-avoiding lattice paths interacting with a wall.We turn to the description of the lattice path model.We first define the class of lattice paths to be considered.Definition 2.5.We call V := (Z + 1  2 ) × Z the vertex set of our lattice paths.An edge is an unordered pair of points e = {x, y}, x, y ∈ V , with euclidean distance d(x, y) = 1.An edge can be horizontal if x = y ± e 1 or vertical if x = y ± e 2 , where e 1 = (1, 0) and e 2 = (0, 1).A self-avoiding lattice path (for short a path in the sequel) joining x ∈ V and y ∈ V is a sequence f 0 , . . ., f n of edges such that: (1) for every i = 0, . . ., n − 1, f i and f i+1 have one common vertex z i ∈ V ; (2) f 0 = {z 0 , z 1 }, f n = {z n , z n+1 } with z 0 = x and z n+1 = y; (3) all vertices z i , i = 0, . . ., n + 1, are distinct.We denote the length of a lattice path γ, that is the number of edges in γ, by |γ|.Given x, y ∈ V , x = y, we call Ω(x, y) the set of all paths joining x and y.Notice that for x = (x 1 , x 2 ) and y = (y 1 , y 2 ), all γ ∈ Ω(x, y) have at least |y 1 − x 1 | horizontal edges.Paths that have exactly |y 1 − x 1 | horizontal edges are in one to one correspondence with random walk trajectories from x − 1 2 e 1 to y + 1 2 e 1 ; see Figure 1.Next, we define the ensemble of lattice paths.Given x, y ∈ V , x = y, we write where β > 0 is the inverse temperature parameter.With slight abuse of notation, when x = ( 1 2 , 0) and y = (L − 1 2 , 0), we write Ω 0,L instead of Ω(x, y) and Z 0,L instead of Z(x, y).Observe that if we restrict to the set Ω0,L of paths with minimal number of horizontal edges then we obtain the following relation with the partition function Z 0,L of the SOS walk with parameter β: where For any γ ∈ Ω 0,L , define where {ε j } is a nonnegative sequence.Define also where Ω +,0 0,L denotes the set of paths γ ∈ Ω 0,L with γ 0. With slight abuse of notation we write again P 0,L and E 0,L (resp.P +,0 0,L and E +,0 0,L ) for the probability and expectation over paths γ ∈ Ω 0,L (resp.γ ∈ Ω +,0 0,L ) with weight e −β|γ| .Remark 2.6 (Alternative definitions of the number of contacts).In analogy with the random walk case one could have considered the number Nj (γ) of horizontal contacts of γ with the j-th level rather than the number N j (γ) defined in (2.14).By horizontal contact with the j-th level here we mean an horizontal edge f ∈ γ at height j.For instance the path in Figure 1 (left) has N 0 (γ) = 7 and N0 (γ) = 3.It is easy to check that all our results apply with no modifications to the potential Φ obtained as in (2.14) with N j replaced by Nj .As we will see in the proof of Theorem 3, our results actually extend to a model where one takes into account also possible contacts of γ with the j-th level occurring outside of the horizontal interval [0, L].

Theorem 4 (Localized phase).
There exist universal constants a, c, β 0 > 0 such that the following holds: For any sequence {ε j }, any integer d 0 and any β β 0 such that one has for all L large enough.
As in Corollary 1 one can immediately infer from Theorem 3 and Theorem 4 the following facts.

Random walks
Here we prove the main results for the random walk model.

3.1.
Proof of the lower bounds.We start with some considerations that apply to both the lower bound (2.4) in Theorem 1 and to (2.9) in Theorem 2. First of all, we can assume that ε j log 2 for each j.Otherwise if ε j * > log 2, one gets immediately exponential growth of the partition function using irreducibility of the walk and the fact that p(0) 1/2 (just consider the trajectory that reaches height j * and sticks there: its weight grows like p(0) L e ε j * L ). Letting one checks immediately that Thus, viewing W L as a matrix and observing that W 1 = P , with P the symmetric matrix P ij = p(i − j)1 i,j≥0 , we have where (•, •) is the scalar product in 2 (Z + ).To see this observe that, if P is the selfadjoint operator P = e V /2 P e V /2 , then (ϕ, e −V (e V P ) L ϕ) = (ψ, ( P ) L ψ).
Since p(0) 1/2, one has that P is non-negative definite, and so is P .Letting µ ψ (dE) denote the spectral measure of P associated with ψ (the total mass is 1 since ψ has unit 2 (Z + )-norm), we have where we used convexity of x L on [0, ∞), for L 1, and Jensen's inequality.
Given an integer d 0, let s(i) = sin(π i+1 d+2 ) and and where we used the symmetry of P .Since .
On the other hand, recalling that 0 ε j log 2 and using exp(−x) Finally, since s(i) c(i + 1)/(d + 1) for 0 i d/2 for some positive constant c, we conclude that (ϕ, P ϕ) Proof of (2.9).Assume by monotonicity that we have equality in (2.8).Then we go back to (3.2), we observe that d 0 s(i) 2 grows linearly in d and that we can assume that d is much larger than σ 2 /d.Then we obtain, for some universal positive constant C, if the value of a in Theorem 2 is chosen large enough.Recalling (3.1) we see that there exist i, j d and positive constants c(d), a such that In turn, using the assumption p(1) > 0, we obtain for some new constant c (d) > 0. This implies the desired lower bound (2.9).
Proof of (2.4).By vertical translation invariance, we can assume that the walk is conditioned to stay non-negative (instead of γ − j) and that the pinning is at height j, i.e. ε i = ε1 i=j .The proof of (2.4) is then essentially identical to the proof of (2.9), once the choice d = 2j is made in the definition of s(•) above.

3.2.
Proof of the upper bound in Theorem 1.Here we prove (2.5).Define Thus, (2.5) can be restated as follows Proposition 3.1.There exists a universal constant b > 0 such that if ε bσ 2 /(j + 1), then uniformly in L, σ 2 , j one has The proof of Proposition 3.1 is divided in two steps.We start with a general lemma for the walk with no wall constraint.This will allow us to cover the region L C(j +1) 2 /σ 2 for any constant C > 0.
Lemma 3.2.There exists c > 0 such that for any L and σ 2 ∈ (0, 1  2 ], any p ∈ P(σ 2 ), for any b < 1/c one has Proof.Writing N as in (2.3), for γ ∈ Ω 0,L one has the expansion where we use e a 1 + 2a for a ∈ (0, 1), and the sum ranges over all possible positions of the internal zeros.We also set x 0 = 0, x n+1 = L.We can assume that σ 2 L is large, since otherwise the statement becomes obvious by estimating N L. Observe that where we write Z u,v = Z 0,v−u .From Proposition B.1 in the appendix one has the local CLT estimates and Z 0,L C −1 σ −1 L −1/2 , for some absolute constant C > 0. It follows that for some absolute constant C > 0. Therefore, It remains to check that for any n 1, for some new absolute constant C > 0 one has: One easily checks that there is C > 0 independent of n, L such that for any fixed values of ξ 1 , . . ., ξ n−1 one has Therefore the sum in (3.7) can be bounded by for some new constant C > 0.
Next, we turn to the proof of the upper bound (2.6) announced in Remark 2.3.It can be restated as follows.
Proof.We first recall the well known bounds (see e.g.[9]): for some constants c i = c i (j, p) > 0 independent of L. From Proposition 3.1 one has Z ε,+,j 0,L 2Z 0,L for all L. By Hölder's inequality, for If k = 3, using (3.14) to estimate P 0,L (γ − j) from above, we obtain with κ = κ(j, p) > 0. We claim that we can bootstrap the bound (3.15) to for some new constant κ = κ (j, p).Once this is achieved, the proposition follows by using the left side in (3.14).
Moreover, neglecting the constraint γ − j in (3.10) one has that Ξ x,y 2Z +,0 x,y .Therefore by (3.14), now with j = 0, one has Ξ x,y κ 0 (y − x) −1 Z x,y , for some κ 0 = κ 0 (p) > 0. Summarizing, we obtain From the local CLT estimates in (3.5) one has for some We need to show that the r.h.s.above is O(1/L).To this end, it suffices to consider the two sums It is not hard to see that the sum in A is always o(L −1 ).On the other hand, the sum in B behaves as 1/L as one easily sees by restricting to either 3.3.Proof of Theorem 2. Let ρ j := (bσ 2 ) −1 (j + 1)ε j .By monotonicity we may assume that ∞ j=0 ρ j = 1.Then, using Jensen's inequality where κ j := ε j /ρ j = bσ 2 /(j + 1).Next, we show that the upper bound in Proposition 3.1 implies if b is small enough, uniformly in j, σ 2 .Notice that this and (3.17) imply the desired estimate (2.11).
To prove (3.18), observe that by a vertical translation, Proposition 3.1 refers to the case where the walk starts and ends at level j, with a wall at zero.Thus, writing Z κ j ,+,j u,v for the partition function of the walk with pinning strength κ j at level j, that starts at j at time u and ends at j at time v, with wall at zero, Proposition 3.1 yields the bound for any 0 u v L. Therefore (3.18) follows by writing where the first sum is over all paths γ 1 0 joining the vertices (0, 0) and (L, 0) that never reach level j, while the second sum is over all paths γ 2 0 joining the same vertices and that touch level j at least once.The first sum is trivially bounded by Z 0,L .The second sum is bounded by summing over the first and last contact with the level j, so that where Ẑ+,0 0,u stands for the sum of w(γ) over all γ 0 joining (0, 0) and (u, j) that never touch height j before the ending point.Similarly, Ẑ+,0 v,L stands for the sum of w(γ) over all γ 0 joining (v, j) and (L, 0) that never touch height j after the starting point.Using (3.19) one obtains immediately that (3.21) is bounded by 2e 2κ j Z 0,L .Since κ j b it follows that (3.18) holds as soon as 1 + 2e 2b 4.

Self-avoiding paths
In this section we prove Theorem 3 and Theorem 4. The strategy follows closely the corresponding arguments in the random walk case.

Lower bound.
Proof of Theorem 4. Recall that if we restrict to the set Ω0,L of paths with minimal number of horizontal edges then we obtain the SOS walk with parameter β; see (2.13).We write Z 0,L , Z +,0 0,L for the usual partition functions of the SOS walk with weight w(γ); see (2.1) and (2.2).By restricting to Ω0,L we may write where the second line follows from the obvious bound Z +,0 0,L Z 0,L and the expectation E +,0,SOS 0,L refers now to the SOS walk.From [8, Eq. (4.8.4)] one has that Ẑ0,L Z 0,L exp (−ce −2β L), for some absolute constant c > 0 and L large enough, where Ẑ0,L is defined in (2.13).Therefore, for some absolute constant c > 0 and L large enough.Moreover, by (3.13) one has Z +,0 0,L c(β)L −1 Z 0,L for some constant c(β) > 0. Finally, the conclusion follows from estimating from below E +,0,SOS 0,L [e Φ ] as in (2.9).

4.2.
Upper bound.We consider the probability measure P 0,L on Ω 0,L defined by the weight e −β|γ| , and write E 0,L for the expectation w.r.t.P 0,L .Call N (γ) = N 0 (γ) the number of contacts with the zero line, as in the definifion (2.14).For lightness of notation, below we set σ 2 := e −β .
We have the following version of Lemma 3.2.Proof.We proceed as in Lemma 3.2.Define χ i (γ) = 1((i + 1 2 , 0) ∈ γ).We have where the sum ranges over all values of the integers 1 where C is a universal constant.Once (4.1) is available, the rest of the proof is exactly as in Lemma 3.2.Given a permutation π = (π(1), . . ., π(n)), call E π the event that in going from ( 12 , 0) to (L − 1 2 , 0), the path γ visits the zeros at {x 1 + 1 2 , . . ., x n + 1 2 } in the order (x π(1) + 1 2 , . . ., x π(n) + 1 2 ).One has Using Proposition B.2 in the appendix it follows that where Ξ k denotes the grand-canonical partition function (the one where the horizontal coordinate of the endpoint of the path is L while the vertical coordinate is free), and we set x π(0) = 0 and x π(n+1) = L. Since for any permutation π one has to establish (4.1) it is sufficient to prove that for some constant C > 0 : As in [8,Eq. (4.8.6)], one has that for some absolute constants C 1 , C 2 > 0, where c(β) > 0 is a constant such that c(β) → ∞ as β → ∞.It follows that for any π: where we define the excess length associated to a permutation π by (π Notice that (π) 0 and (π) = 0 iff π is the identity π(i) ≡ i.To conclude, we show that where δ(β) → 0 when β → ∞.The statement (4.2) can be obtained by induction over n, as follows.Fix x 0 = 0 and x n+1 = L and let x 1 , . . ., x n be arbitrary integers satisfying x i = 0 and Here we do not assume that x 1 > 0. Let φ(n) be defined as the sum in (4.2) for this choice of points {x i }.We claim that where x − 1 := max(−x 1 , 0) is the negative part of x 1 .Clearly, (4.3) is sufficient to prove (4.2) which corresponds to the case x 1 > 0. To prove (4.3) notice that for n = 1 one has π(1) = 1 and (π) = 2x − 1 so that (4.3) is satisfied in this case.For k ∈ N and δ > 0, assume φ(k) (1 + δ) k e −c(β)x − 1 and consider the case n = k + 1. Decomposing along the value of π(1) one has By the inductive assumption one has: if j is such that x j > 0, then while if j is such that x j < 0 (and thus x 1 < 0), then To conclude observe that if x 1 > 0 (and thus x j > 0 for all j 1), then one has which is bounded by 2δ(1 + δ) k e −c(β)x − 1 .This implies the claim (4.3).
We turn to the proof of Theorem 3. We need a version of Proposition 3.1 for lattice paths.For integers j 0, define where the condition γ −j means that all vertices of γ have vertical coordinate at least −j.For the purpose of the recursive argument we have to consider also the number of external contacts of the path with the zero line.Namely, for any γ ∈ Ω 0,L , define If N (γ) denotes the number of internal contacts with level zero as in (2.14), for ε > 0 we write where ε = bσ 2 /(j + 1).Lemma 4.3.For any δ > 0, there exist constants b 0 > 0, β 0 > 0 such that H 1 (b, δ) holds for all b b 0 and β β 0 .
From Corollary A.5 we know that E 0,L e 2εNext(γ) 1 + u β for some constant u β → 0 as β → ∞.Thus, using Schwarz' inequality we have It remains to take b, β such that (1 The next proposition establishes that H n (δ, b) holds in fact for all n ∈ N if we take e.g.δ = 1, b small enough, and β large enough.Proposition 4.4.There exist constants b > 0, β 0 > 0 such that if ε b e −β /(j + 1), then uniformly in L, j and β β 0 one has Proof.We repeat the very same argument of Lemma 3.4.The variable ξ(γ) is now defined as the horizontal coordinate of the last contact of γ with the zero line in the interval [0, L/2], while η(γ) denotes the horizontal coordinate of the first contact with the zero line in the interval (L/2, L].Equation (3.11) now becomes: for any L ∈ (L n , L n+1 ], where Ξ ε x,y := γ∈Ω((x,0),(y,0)) Note that the above sum includes the contribution of possible external zeros.This comes from the event that the path γ intersects the zero line outside of the interval [x, y].Moreover, in (4.6) we have used the inductive assumption H n (δ, b).Notice that due to the possible presence of overhangs in the path γ it is crucial to take into account the external contacts N ext in order to exploit the induction.From Corollary A.5 in the appendix we know that Ξ ε x,y (1 + u β )Ξ 0 x,y for some constant u β → 0 as β → ∞.Moreover, Proposition A.2 in the appendix shows that Ξ 0 x,y x,y , where Ξ 0 x,y denotes the sum Ξ 0 x,y in (4.7) restricted to paths that are regular at the endpoints, see Definition A.1.From Corollary A.7 one has Z 0,x (1 + u β ) Z 0,x and Z y,L (1 + u β ) Z y,L where Z u,v denotes the partition function Z u,v restricted to paths that start and end with horizontal edges and that are regular at the points x = u + 1 and x = v − 1.Notice that if γ 1 is a path appearing in Z 0,x , γ 2 is a path from Ξ 0 x,y , and γ 3 is a path from Z y,L , then the composition γ 1 • γ 2 • γ 3 defines a valid path from 0 to L. That is, the above restrictions allow us to avoid complications due to the self-avoiding constraint when we reconstruct the global partition function Z 0,L .
From (4.6) we obtain, for some constant Define γ max ( ) as the maximal vertical height of γ at points with horizontal coordinate .Then it is not hard to see that (4.8) implies As in the proof of Proposition 3.1, it remains to show that P 0,L (γ max (L/2) −j) 1−δ 0 for some δ 0 > 0, uniformly in L L 1 = C(j + 1) 2 e β , j 0 and β > β 0 .This follows from Proposition B.2 (b) in the appendix.
Proof of Theorem 3. We proceed exactly as in (3.17).Thus, where κ j := ε j /ρ j = bσ 2 /(j + 1), and ∞ j=0 ρ j = 1.The estimate (3.18) now takes the form γ∈Ω +,0 0,L e −β|γ| e κ j N j (γ) 4Z 0,L . ( To prove (4.9), thanks to Proposition 4.4 the same reasoning as in (3.21) can be applied with minor modifications.The only difference is that to conclude one needs to handle the partition functions in (3.21) with some care in order to restore the final partition function Z 0,L .More precisely, fix u, v ∈ Z, with 0 u v L; let Ẑ+,0 0,u denote the partition function corresponding to paths j γ 0 from ( 1 2 , 0) to (u + 1 2 , j) that never touch level j before the endpoint; similarly, let Ẑ+,0 v,L denote the partition function corresponding to paths j γ 0 from (v − 1 2 , j) to (L − 1 2 , 0) that never touch level j after the staring point.Then, reasoning as in (3.20) and (3.21), the sum in (4.9) is bounded above by where Z ε,+,j u,v = Z ε,+,j 0,v−u is defined as in (4.5) above.Note that for this estimate to hold it is crucial that Z ε,+,j u,v takes into account the interaction with external zeros.By Proposition 4.4 one has Z ε,+,j u,v 2Z u,v .Moreover, let Z +,0 0,u , Z +,0 v,L denote the partition functions Ẑ+,0 0,u and Ẑ+,0 v,L respectively defined with the restriction that the path is regular at the endpoints, see Definition A.1.Then, from Proposition A.2 in the appendix one has that Ẑ+,0 , where Z u,v denotes the partition function Z u,v restricted to paths that start and end with horizontal edges and that are regular at the points x = u + 1 and , for some v β → 0 as β → ∞ As discussed before Eq.(4.8), the regularity constraint at the endpoints together with the restriction to horizontal edges allows us to reconstruct the partition function Z 0,L as an upper bound so that, taking β large enough, one can conclude in the same way as in the proof of (3.18).

Appendix A.
In this first appendix we prove few technical results which, roughly speaking, say that, even in the presence of a wall, the lattice path ensemble for β large is likely to intersect a given vertical line only once, not to make excursions to the left of its starting point (or to the right of the final point) and to start and to end with a horizontal bond.
These very intuitive results are useful when trying to concatenate together different pieces of the path in our recursive method.
Consider the lattice path model defined in Section 2.2.We introduce a bit more notation.Given H ∈ Z we will denote by Ω H 0,L the space of self-avoiding lattice paths γ connecting (1/2, 0) to (L − 1/2, H).The corresponding ensemble will be denoted by P H 0,L .If H = 0 we will simply write (Ω 0,L , P 0,L ) as usual.The vertical line through the point (u, 0), u ∈ Z, will be denoted by L u and the cardinality of a finite set S will be denoted by |S|.We write γ − j if the height of γ ∈ Ω H 0,L is everywhere at least −j.Below, j is always a non-negative integer.
L is regular at both 0 and L we say that γ is regular at the endpoints.As an example, both paths in Figure 1 are regular at the endpoints, the left path is regular at u = 2, while the right path is not.
Proposition A.2.We have For simplicity we only treat the case u = 0 but the same strategy with minor modifications works for other values of u.Let PH 0,L (•) = P H 0,L (• | γ ∩ L L = ∅) and let A denote the event that γ ∈ Ω H 0,L is not regular at 0. We will first show that lim Later on we will show that (A.2) implies the same bound for P H 0,L .To prove (A.2) fix δ 1/10 (independent of β) and let G L be the event that there exists y ∈ [L/2 − δ 2 L 1−δ , L/2 + δ 2 L 1−δ ] ∩ Z such that γ ∩ L y = (y, H y ) and H y is within δ 2 L 1−δ from the average height Hy := Hy/L.Lemma A.3.For all β large enough and uniformly in j 0, |H| 3L/2 and L, for some constant c = c(δ, β) with lim β→∞ c(δ, β) = +∞.The same bound applies to Proof of the Lemma.We prove the lemma for PH 0,L but the same arguments apply to Using [8,Section 4.14] the probability that there exists y ∈ [L/2−δ 2 L 1−δ , L/2+δ 2 L 1−δ ]∩ Z such that γ ∩ L y contains a point whose height differs from Hy by more than δ 2 L 1−δ is smaller than e −cL 1−2δ for some constant c = c(δ).If for all y ∈ [L/2 − δ 2 L 1−δ , L/2 + δ 2 L 1−δ ] ∩ Z the set γ ∩ L y is not a singleton and it is contained in the interval [ Hy ± the path γ has length at least 10δ 2 L 1−δ i.e. an excess length (w.r.t. to its minimal length) of at least (6 − 2H/L)δ 2 L 1−δ 3δ 2 L 1−δ .Therefore a Peierls argument shows that the above event has probability not larger than e −c βL 1−δ for some constant c = c (δ).In conclusion, by renaming the constants if necessary, PH 0,L (G c L ) e −cL 1−2δ for some constant c = c(δ, β) diverging as β → +∞.
We conclude with a rough lower bound on PH 0,L (γ − j) of the form for some constant c = c (δ).To prove this, let = L 1−3δ with δ < 1/6.We can restrict the sum in the numerator above to paths which, while staying above −j and never intersecting L L , first go straight to the point (0, ), then reach the point (L, H + ) and finally go straight to the point (L, H).Since L 1/2 , [8, ?] implies that the extra constraint of staying above level −j is irrelevant for this restricted sum which is therefore greater than e.g. e −3β γ∩L L =∅ e −β|γ| .Hence the claimed bound.We return to the proof of (A.2) let We claim that p L p L/2 + exp(−cL Notice that the law of the part of the path γ joining the origin to (ξ − 1/2, H ξ ) is exactly PH 0,ξ .Hence, using Lemma A.3 and (A.5), we get PH i.e. (A.4).To finish the proof of (A.2) it is sufficient to observe that, for any fixed L 0 , because the event A forces the path to have an excess length (w.r.t. to the minimal lenght) of at least 2. Hence (A.4) implies that, for any L 0 , lim sup and (A.2) follows.
We finally observe that (A.1) follows at once from (A.2).Fix some large L 0 independent of β.For L L 0 a simple Peierls argument shows that For L > L 0 write as before In the first term in the r.h.s. the path to the left of L ξ has exactly the distribution PH 0,ξ .Hence its β → ∞ limit is zero by (A.2).The second term in the r.h.s. is smaller than e −cL 1−2δ e −cL 1−2δ 0 by the previous lemma.Since L 0 was arbitrary the result follows.This ends the proof of Proposition A.2.
Corollary A.5.Let N ext (γ) denote the external zeros as defined in (4.4).Then, for any fixed constant a > 0: Suppose now that L 1−δ .The argument leading to (A.4), with we can write and that, using the assumption p ∈ P(σ 2 ), for all |t| πσ n .Above we used the formula Var(Z) = 1 2 E((Z − Z ) 2 ) where Z, Z are two independent copies of the same random variable to get In conclusion |f n (t)| e −c 0 t 2 /8 and for some constant C depending on c 0 .To prove a lower bound we use to write where we used the assumptions E(X 1 ) = 0 and E(|X 1 | 3 ) cσ 2 .Thus, if σ n 1 and by choosing δ small enough independent of σ 2 , we obtain If instead σ n 1 we simply write . Equation (B.1) follows if we observe that (without loss of generality we assume L even) We turn to the proof of part (b).With the previous notation we can write (n = L/2) Before proving the lemma we recall some key results from [8].For any γ ∈ O L let h L (γ) be the height of its final point.Let also σ 2 L , f L (t) be the variance of h L (γ) and the characteristic function of h L (γ)/σ L respectively in the ensemble P L .
Lemma B.3 ([8, Sections 4.9,4.10,4.10.20,4.10.29]).There exists β 0 > 0 such that, for all β β 0 , the following holds.Next we lower bound Z 0,L by Z reg 0,L .Any γ ∈ Ω reg 0,L can be seen as formed by two paths γ i , i = 1, 2, the first one starting from the origin and running forward and the second one starting from (L, 0) and running backward, until they hit L n at the same height.These two paths are i.i.d with law Pn (γ).Thus We end with a simple lemma that allows one to bound canonical probabilities with their grand canonical counterpart.We use the notation introduced so far.

6 )
Proof of the Corollary.We may write N ext = N − (γ) + N + (γ), where N + (γ) (resp.N − (γ)) denotes the number of zeros to the right (resp.left) of the interval [0, L].By Schwarz' inequality and using symmetry it suffices to prove the statement (A.6) with N ext replaced by N − (γ).Clearly N − (γ) |X(γ)|.The statement then follows at once from Proposition A.4. Proof of Proposition A.4. Fix δ 1/10 and assume first that L 1−δ .In this case, using (A.3), it suffices to prove the required estimate for the unconditioned probability.If by traveling along the path the last intersection with the column L 1 is at height Y with |Y | then the part of the path between the origin and the vertex (1, Y ) has an excess length of at least .A standard Peierls argument shows that the unconditioned probability of such an event is bounded from above by e −m β with lim β→∞ m β = +∞.If instead |Y | we can appeal to the large deviation bounds in [8, Section 4] to get the same result.

Lemma B. 4 .
Fix k ∈ N, L k and let A ⊂ Ω 0,L depend only on the first k edges of the path γ.If lim β→∞ sup L k P L (A) = 0 then the same holds for the canonical measure P 0,L .Proof.Fix a large constant C. If L Ce β then it is immediate to check that, as β → ∞, the measure P 0,L is concentrated over paths which are flat for the first k steps as is the case for the grand canonical measure, so the two marginals on the first k steps are essentially identical.If instead L Ce β we argue as in part (b) of Proposition B.2 to write (with n = L ) Pn ⊗ Pn (A | h n (γ 1 ) = h n (γ 2 )) dt | fn (t)| πσn −πσn dt | fn (t)| 2 Pn (A) ( δ = δ(C) tends to zero as C → +∞.Finally we observe that lim