Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation $t$ (and a horizontal separation of the same order), the maximum transversal fluctuation has order $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$, $z\in [1,2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3}\big(\log t^{-1}\big)^{2/3}$. In this way, we identify exponent pairs of $(2/3,1/3)$ and $(1/3,2/3)$ in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [10,11,9] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.


Introduction
In 1986, Kardar, Parisi, and Zhang [13] predicted universal scaling behaviour for many planar random growth processes, including first and last passage percolation as well as corner growth processes, though rigorous validation has been subsequently provided for only a handful of them.In such models, fluctuation in the direction of growth is governed by an exponent of one-third, with this fluctuation enduring on a scale governed by an exponent of two-thirds in the orthogonal, or transversal, direction.
Poissonian last passage percolation illustrates these effects.We will define it shortly, since it is our object of study; briefly, the model specifies a growth process whose height at a given moment is the maximum number of points (or the energy) obtainable in a directed path through a planar Poisson point process.Baik, Deift and Johansson [1] established the n 1/3 -order fluctuation of the maximum number of Poisson points on an increasing path from (0, 0) to (n, n), deriving the GUE Tracy-Widom distributional limit of the scaled energy.Later Johansson [12] proved the transversal fluctuation exponent of two-thirds in this model.These are exactly solvable models, for which certain exact distributional formulas are available, and the derivations of these formulas typically employ deep machinery from algebraic combinatorics or random matrix theory.It is interesting to study geometric properties of universal KPZ objects by approaches that, while they are reliant on certain integrable inputs, are probabilistic in flavour: for example, [4], [2] and [3] are recent results and applications concerning geometric properties of last passage percolation paths.
It is rigorously understood, then, that last passage percolation paths experience fluctuation in their energy and transversal fluctuation governed by scaling exponents of one-third and twothirds.It is very natural to view such paths via the lens of scaled coordinates, in which transversal fluctuation and path energy each has unit order.We will be more precise very shortly, when suitable notation has been introduced, but for now we mention that our aim in this article is to refine rigorous understanding of the magnitude and geometry of fluctuation in last passage percolation paths.We shall call the scaled geodesics polymers, and refer to the scaled energy as weight.We will see that polylogarithmic corrections to the scaled laws implied by the exponents of one-third and two-thirds arise when we consider natural geometric problems concerning the weights and the maximum fluctuation among polymers in a unit order region.The techniques for verifying our claims will employ geometric and probabilistic tools rather than principally integrable ones, since problems involving maxima as both endpoints of a last passage percolation path are varied are not usually amenable to integrable techniques.
1.1.Model definition and main results.Let Π be a homogeneous rate one Poisson point process (PPP) on R 2 .We introduce a partial order on R 2 : (x 1 , y 1 ) (x 2 , y 2 ) if and only if x 1 ≤ x 2 and y 1 ≤ y 2 .For u v, u, v ∈ R 2 , an increasing path γ from u to v is a piecewise affine path, viewed as a subset of R 2 , that joins points u = u 0 u 1 u 2 . . .u k = v such that u i ∈ Π for i ∈ 1, k − 1 .Here and later, a, b for a, b ∈ Z with a ≤ b denotes the integer interval {a, • • • , b}.Also let |γ| denote the energy of γ, namely the number of points in Π \ {v} that lie on γ; (the last vertex is excluded from the definition of energy so that the sum of the energies of two paths equals the energy of the concatenated path, as we will see in Section 3.1).Then we define the last passage time from u to v, denoted by X v u , to be the maximum of |γ| as γ varies over all increasing paths from u to v. Any such maximizing path is called a geodesic.There may be several such, but if Γ v u denotes any one of them, we have Note that, in this notation, the starting and ending points of the geodesic, u and v, are assigned subscript and superscript placements.We will often use this convention, including in the case of the scaled coordinates that we will introduce momentarily.
When u v, any geodesic from u to v may be viewed as a function of its horizontal coordinate, since it contains a vertical line segment with probability zero.The operations of maximum and minimum may be applied to any pair of such geodesics, and the results are also geodesics.For this reason, we may speak unambiguously of Γ ←;v u , the uppermost geodesic between u and v, and of Γ →;v u , the lowermost geodesic between u and v. (The notation ← and → is compatible with these two paths being equally well described as the leftmost and rightmost geodesics.This choice of notation also anticipates the form of these paths when viewed in the scaled coordinates that we are about to introduce.)When the endpoints are (0, 0) and (n, n), we will call these geodesics Γ ← n and Γ → n .
1.1.1.Introducing scaled coordinates.We rotate the plane about the origin counterclockwise by 45 degrees, squeeze the vertical coordinate by a factor 2 1/2 n and the horizontal one by 2 1/2 n 2/3 , thus setting The horizontal line at vertical coordinate t is the image under T n of the anti-diagonal line through (nt, nt).It is easy to see that, for (x, t) ∈ R 2 , T −1 n (x, t) = (nt + xn 2/3 , nt − xn 2/3 ).Paths that are the image of geodesics under T n will be called polymers; we might say n-polymers, but the suppressed parameter will always be n.Geodesics from (0, 0) to (n, n) transform to polymers (0, 0) to (0, 1). Figure 1 depicts a geodesic Γ and its image polymer ρ.The polymer between planar points u and v that is the image of the uppermost geodesic given the preimage endpoints will be denoted by ρ ←;v n;u , and, naturally enough, called the leftmost polymer from u to v. The rightmost polymer from u to v is the image of the corresponding lowermost geodesic and will be denoted by ρ →;v n;u .The simpler notation ρ ← n and ρ → n will be adopted when u = (0, 0) and v = (0, 1).When T −1 n (x 2 , t 2 ), we will, when it is convenient, regard any polymer ρ from u to v as a function of its vertical coordinate: that is, for t ∈ [t 1 , t 2 ], ρ(t) will denote the unique point such that (ρ(t), t) ∈ ρ. (This definition makes sense since an increasing path can intersect any anti-diagonal at most once.)We regard the vertical coordinate as time, as the t-notation suggests, and will sometimes refer to the interval [t 1 , t 2 ] as the lifetime of the polymer.In particular, when t 1 = 0, t 2 = 1, writing C[0, 1] for the space of continuous real-valued functions on [0, 1] (equipped for later purposes with the topology of uniform convergence), we may thus view ρ = {ρ(t)} t∈[0,1] as an element of C[0, 1].
. Indeed, we will write ; this condition ensures that polymers exist between the endpoints u and v.
The first of our three main results shows that polymers, viewed as functions of the vertical coordinate, enjoy modulus of continuity of order t 2/3 log t −1 1/3 .
There exists a constant C > 0 such that, for the weak limit ρ ← * of any weakly converging subsequence of {ρ ← n } n∈N , almost surely, The same result holds for the rightmost polymer.
(0, 0) Note that the constant C does not depend on the choice of the weakly converging subsequence.
The exponent pair (2/3, 1/3) for power law and polylogarithmic correction is thus demonstrated to hold in an upper bound on polymer fluctuation.We believe that a lower bound holds as well, in the sense that the limit infimum counterpart to (3) is positive.A polymer is an object specified by a global constraint, and it by no means clearly enjoys independence properties as it traverses disjoint regions, even though the underlying Poisson randomness does.In order to demonstrate the polymer fluctuation lower bound, this subtlety would have to be addressed.We choose instead to demonstrate that the exponent pair (2/3, 1/3) describes polymer fluctuation by proving a lower bound of this form for the maximum fluctuation witnessed among a natural class of short polymers in a unit region.This alternative formulation offers a greater supply of independent randomness.Indeed, we now specify a notion of maximum transversal fluctuation over a collection of short polymers.Fix any two points u = (x 1 , t 1 ), v = (x 2 , t 2 ) such that t 2 > t 1 .Let Φ v n;u denote the set of all polymers ρ from u to v. Let v u denote the planar line segment that joins u and v; extending an abuse of notation that we have already made, we write v u (t) for the unique point such that Then, for any polymer ρ, the transversal fluctuation TF(ρ) of ρ is specified to be and the transversal fluctuation between the points u and v to be Also, let denote the reciprocal of the slope of the interpolating line.Since Now fix some large constant ψ > 0.Then, for any fixed parameter t ∈ (0, 1] and any n ∈ N, n > ψ 3 , we define the set of admissible endpoint pairs AdEndPair n,ψ (t) := ((x 1 , t 1 ), (x 2 , t 2 )) : Since Recalling the notation at the start of Subsection 1.1.2,we thus have (x 1 , t 1 ) n (x 2 , t 2 ), so that polymers do exist between such endpoint pairs.
We then define so that MTF n (t) is the maximum transversal fluctuation over polymers between all endpoint pairs at vertical distance at most t such that the slope of the interpolating line segment is bounded away from being horizontal; (we suppress the parameter ψ in the notation).Our second theorem demonstrates that the exponent pair (2/3, 1/3) governs this maximum traversal fluctuation.
Theorem 1.2.There exist ψ-determined constants 0 < c < C < ∞ such that lim inf 1.1.3.Scaled energies are called weights.It is natural to scale the energy of a geodesic when we view the geodesic as a polymer after scaling.Scaled energy will be called weight and specified so that it is of unit order for polymers that cross unit-order distances.For t 1 < t 2 , let t 1,2 denote t 2 − t 1 ; (this is a notation that we will often use).Let (x, t 1 ), (y, t 2 ) ∈ R 2 be such that |x − y| < t 1,2 n 1/3 .
(This condition ensures that (x, t 1 ) n (y, t 2 ), so that polymers exist between this pair of points.)Since T −1 n ((x, t 1 )) = (nt 1 + xn 2/3 , nt 1 − xn 2/3 ) and T −1 n ((y, t 2 )) = (nt 2 + yn 2/3 , nt 2 − yn 2/3 ), it is natural to define the scaled energies, which we call weights, in the following way.Define Because of translation invariance of the underlying Poisson point process, t 1,2 is a far more relevant parameter than t 1 or t 2 .The notation on the left-hand side of ( 8) is characteristic of our presentation in this article: a scaled object is being denoted, with planar points (•, •) in the subscript and superscript indicating starting and ending points.
1.1.4.A continuous modification of the weight function.For the statement of our third theorem, we prefer to make an adjustment to the polymer weight to cope with a minor problem concerning discontinuity of geodesic energy under endpoint perturbation.For n ∈ N, define Observe that X n (t) is integer-valued, non-decreasing, right continuous and has almost surely a finite number of jump discontinuities.Let d 0 = 1 and d m = 2. Record in increasing order the points of discontinuity of X n as a list d 1 , d 2 , • • • , d m−1 .We specify a modified and continuous form of the function X n by linearly interpolating it between these points of discontinuity, setting Because almost surely no two points in a planar Poisson point process share either their horizontal or vertical coordinate, X n (d i+1 ) − X n (d i ) = 1 for all i.Thus, for all t ∈ [1, 2], Now define the modified weight function Wgt n : [1, 2] → R for polymers from (0, 1) to (•, 1): Because of (9), By construction, Wgt n sending t ∈ [1,2] to Wgt n (t) is an element of C [1,2], the space of continuous functions on [1,2]; (similarly to before, this space will be equipped with the topology of uniform convergence).
Our third main result demonstrates that the exponent pair (1/3, 2/3) offers a description of the modulus of continuity of polymer weight when one endpoint is varied vertically.
There exist constants 0 < c < C < ∞ such that, for the weak limit Wgt * of any weakly converging subsequence of {Wgt n } n∈N , almost surely ≤ lim sup Note that, as in Theorem 1.1, the constants c and C do not depend on the choice of weak limit point or converging subsequence.

1.2.
A few words about the proofs.The main ingredients in the proofs of Theorem 1.1 and Theorem 1.2 are the estimates from integrable probability assembled in Section 2 and a polymer ordering property elaborated in Lemma 3.2 that propagates control on polymer fluctuation among polymers whose endpoints lie in a discrete mesh to all polymers in the region of this mesh.The basic tools in the proof of the upper bound in Theorem 1.3 and that of Proposition 1.4 are surgical techniques and comparisons of the weights of polymers, and are reminiscent of the techniques developed and extensively used in [4] and [2].1.3.Phase separation and KPZ.Certain random models manifest the scaling exponents of KPZ universality and some of its qualitative features, without exhibiting the richness of behaviour of models in this class.For example, the least convex majorant of the stochastic process R → R : x → B(x) − t −1 x 2 is comprised of planar line segments, or facets, the largest of which in a compact region has length of order t 2/3+o(1) when t > 0 is high; and the typical deviation of the process from its majorant scales as t 1/3+o (1) .Some such models form a testing ground for KPZ conjectures.Phase separation concerns the form of the boundary of a droplet of one substance suspended in another.When supercritical bond percolation on Z 2 is conditioned on the cluster (or droplet) containing the origin being finite and large, namely of finite size at least n 2 , with n high, the interface at the boundary of this cluster is expected to exhibit KPZ scaling characteristics, with the scaling parameter n playing a comparable role to t in the preceding example.Indeed, in [10,11,9], a surrogate of this interface, expressed in terms of the random cluster model, was investigated.The maximum length of the facets that comprise the boundary of the interface's convex hull was proved to typically have the order n 2/3 log n 1/3 , while the maximum local roughness, namely the maximum distance from a point on the interface to the convex hull boundary, was shown to be of the order of n 1/3 log n 2/3 .
Viewed in this light, the present article validates for the KPZ universality class the implied predictions: that exponent pairs of (1/3, 2, 3) and (2/3, 1/3) for power-law and logarthmic-power govern maximal polymer weight change under vertical endpoint displacement and maximal transversal polymer fluctuation.
In a natural sense, these two exponent pairs are accompanied by a third, namely (1/2, 1/2), for interface regularity.In the example of parabolically curved Brownian motion, x → B(x) − x 2 t −1 , the modulus of continuity of the process on [−1, 1] is easily seen to have the form s 1/2 log s −1 1/2 , up to a random constant, and uniformly in t ≥ 1.In KPZ, this assertion finds a counterpart when it is made for the Airy 2 process, which offers a limiting description in scaled coordinates of the weight of polymers of given lifetime with first endpoint fixed.This assertion has been proved in [7,Theorem 1.11(1)].Recently, for a very broad class of initial data, the polymer weight profile was shown in [8,Theorem 1.2] to have a modulus of continuity of the order of s 1/2 log s −1 2/3 , uniformly in the scaling parameter and the initial condition.
1.4.Organization.We continue with two sections that offer basic general tools.The first, Section 2, provides useful estimates available from the integrable probability literature.Then, in Section 3, we state and prove the polymer ordering lemmas and some other basic results, which are essential tools in the proofs of the main theorems.
The remaining four sections, 4 -7, contain the main proofs.Consecutively, these sections are devoted to proving: • the polymer Hölder continuity upper bound Theorem 1.1; • the modulus of continuity for maximum transversal fluctuation over short polymers, Theorem 1.2, subject to assuming Proposition 1.4; • Hölder continuity for the polymer weight profile, Theorem 1.3; • and the lower bound on transversal polymer fluctuation, Proposition 1.4.
We will stick to scaled coordinates in the results' statements and, except in Section 2, in their proofs.A bridge between scaled coordinates and the original ones is offered in this next section, in whose proofs we use the scaling map T n from (2) and weight function W from (8) to transfer unscaled results to their scaled counterparts.

Scalings and estimates from integrable probability
In this section, we assemble some results from integrable probability.Most of these results were derived in terms of unscaled coordinates in [4] and [2].Point-to-point estimates of last passage percolation geodesics were used crucially in [4] to resolve the "slow-bond" conjecture, and in [2] to show the coalescence of nearby geodesics, and those estimates will be crucially employed in this paper as well.We state the results in scaled coordinates, and the proofs detail how to obtain these statements from their unscaled versions available in the literature.In going from the unscaled to scaled coordinates, we shall use the definitions of the scaling map in (2) and the weight in (8).First we observe some simple relations between the different scaled versions of these quantities that will be used in the proofs of the theorems in this section.
The scaling principle.Because of translation invariance and the definition (2), it is easy to see that for any x, y, t The same statement holds for the rightmost polymers as well.Here and throughout d = denotes that the two random variables on either side have the same distribution.We will sometimes call the displayed assertion the scaling principle.
Also by translation invariance and the definition of weight in (8), it follows that Boldface notation for applying results.In our proofs, we will naturally often be applying tools such as those stated in this section.Sometimes the notation of the tool and of the context of the application will be in conflict.To alleviate this conflict, we will use boldface notation when we specify the values of the parameters of a given tool in terms of quantities in the context of the application.We will first use this notational device shortly, in one of the upcoming proofs.
The next theorem was proved in [1].
n;(0,0) ⇒ F T W , where the convergence is in distribution and F T W denotes the GUE Tracy-Widom distribution.
For a definition of the GUE Tracy-Widom distribution, also called the F 2 distribution, see [1].Moderate deviation inequalities for this centred and scaled polymer weight will be important.Such inequalities follow immediately from [14, Theorem 1.3], [15, Theorem 1.2] and (14).These are essential inequalities, used repeatedly in this paper.In fact, it should be possible to recover the theorems of this paper for other integrable models for which such moderate deviation estimates are known.
Theorem 2.2.There exist positive constants c, s 0 and n 0 such that, for all t 1 < t 2 with nt 1,2 > n 0 and s > s 0 , n;(0,t 1 ) ≥ s ≤ e −cs 3/2 , and Also, we shall need not just tail bounds for weights of point to point polymers, but uniform tail bounds on polymer weights whose endpoints vary over fixed unit order intervals.The unscaled version of this theorem follows from [4, Propositions 10.1 and 10.5].

Theorem 2.3. There exist
1,2 and I and J intervals of length at most t Proof.First we prove the theorem when t 1 = 0 and t 2 = 1 by invoking the unscaled version of this theorem from [4].At the end we prove Theorem 2.3 for general ).If S u,v denotes the slope of the line segment joining u and v, then |x − y| < 2 −1 n ensures that 3 −1 < S u,v < 3.Then, using the first order estimates (see [4,Corollary 9.1]) and a simple binomial expansion giving 2 .Hence, using the definition of the weight function in (8), for all s ≥ 6C 2 , W (y,1) Following the proofs of Proposition 10.1 and 10.5 of [4] verbatim, and using the above bound in (15) in place of Corollary 9.1 of [4], one thus has for all n, s large enough, Thus, for n large enough, and I and J intervals of at most unit length contained in the interval of length 2C −1 0 s 1/4 n 1/6 centred at the origin, We now make a first use of the boldface notation for applying results specified at the beginning of Section 2. For general 1,2 J and s = s in (16).Recall that the boldface variables are those of Theorem 2.3 and that these are written in terms of non-boldface parameters specified by the present context.
From the hypothesis of Theorem 2.3, I and J are intervals of at most unit length contained in [−n 1/6 , n 1/6 ].Thus, applying (16) and using the scaling principle ( 14), we get Theorem 2.3.
The following lower bound on the tail of the polymer weight distribution follows from [14, Theorem 1.3] and (14).
Theorem 2.4.There exist constants c 2 , s 0 , n 0 > 0 such that, for all t 1 < t 2 with nt 1,2 > n 0 and s > s 0 , Moving to unscaled coordinates, the transversal fluctuations for paths between (0, 0) and (n, n) around the interpolating line joining the two points were shown to be n 2/3+o (1) with high probability in [12].More precise estimates were established in [4].However, the fluctuation of the geodesic at the point (r, r) for any r ≤ n is only of the order r 2/3 .This is the content of the next theorem which in essence is the scaled version of [2, Theorem 2] adapted for Poissonian LPP.Recall that, for u, v ∈ R 2 , Φ v n;u is the set of all polymers from u to v, and v u is the straight line joining u and v. Theorem 2.5.There exist positive constants n 0 , s 1 , c such that for all x, y, t Here a ∧ b denotes min{a, b}.Proof of Theorem 2.5.First we prove the theorem when t 1 = 0, t 2 = 1, x = 0. Observe that in this case it is enough to bound the probabilities of the events ρ ←;(y,1) , and use a union bound to obtain (17).We first prove an upper bound for the probability of the first of these two events.Also, first assume that t ∈ [0, 2 −1 ].To prove the bound in this case, we move to unscaled coordinates, and use [2, Theorem 2].
To this end, let Γ := Γ ←;(n+yn 2/3 ,n−yn 2/3 ) (0,0) be the leftmost geodesic, and S the straight line from (0, 0) to (n + yn 2/3 , n − yn 2/3 ).For r ∈ [0, n + yn 2/3 ], let Γ(r) and S(r) be such that (r, Γ(r)) ∈ Γ and (r, S(r)) ∈ S. Now, for r = nt, where r is such that the anti-diagonal line passing through (r, r) intersects S at (r , S(r )).The last inclusion follows from the definition of the scaling map Thus it is enough to bound the probability of the event C. ].This gives that, for some positive constants n 0 , r 0 , s 0 , and for n ≥ n 0 , r ≥ r 0 and s ≥ s 0 , However, observe that (20) holds only when r ≥ r 0 .Now assume r ≤ r 0 , so that r ≤ r 0 , where r 0 = 2r 0 .Let the anti-diagonal passing through (r, r) intersect the geodesic Γ at v and the line S at w.
The following theorem bounds the transversal fluctuation of polymers; (recall the definitions in (4) and ( 5)).The theorem essentially follows from [4,Theorem 11.1]; however, we replace the exponent in the upper bound with its optimal value.Theorem 2.6.There exist positive constants c, n 0 and k 0 such that, for t Proof.Because of (5), it is enough to bound the probabilities of the events TF ρ ←;(0,t) n;(0,0) ≥ kt 2/3 and TF ρ →;(0,t) n;(0,0) ≥ kt 2/3 and use a union bound.We bound only the first event, the arguments for the second event being the same.Then, as in the proof of Theorem 2.5, going to the unscaled coordinates, and defining Γ = Γ ←;(nt,nt) (0,0) , it is enough to show that From Theorem 2.5, it is easy to see that there exist constants c > 0 and n 0 , k 0 > 0 such that, for all k > k 0 and nt ≥ n 0 , Using the above bound in place of [4,Lemma 11.3], and following the rest of the proof of [4,Theorem 11.1] verbatim, we get (22).

Basic tools
Fundamental facts about ordering and concatenation of polymers will be used repeatedly in the proofs of the main theorems.
3.1.Polymer concatenation and superadditivity of weights.Let n ∈ N and (x, t 1 ), (y, t 2 ) ∈ R 2 with t 1 < t 2 and |x − y| < n 1/3 (t 2 − t 1 ).(This condition ensures that (x, t 1 ) n (y, t 2 ), see Subsection 1.1.2.)Let u = T −1 n (x, t 1 ) and v = T −1 n (y, t 2 ) and let ζ be an increasing path from u to v. Let γ = T n (ζ).We call γ an n-path.We shall often consider γ as a subset of R 2 , and call (x, t 1 ) its starting point and (y, t 2 ) its ending point.Moreover, similarly to the definition of the weight of a polymer in (8), we define the weight of an n-path as where |ζ| denotes the energy of ζ, that is, the number of points in Π \ {v} that lie on ζ.Now, let (x, t 1 ), (y, t 2 ), (z, t 3 ) ∈ R 2 be such that t 1 < t 2 < t 3 , |x − y| < n 1/3 (t 2 − t 1 ) and |y − z| < n 1/3 (t 3 − t 2 ), so that there exist polymers from (x, t 1 ) to (y, t 2 ); and from (y, t 2 ) to (z, t 3 ).Let ρ 1 be any polymer from (x, t 1 ) to (y, t 2 ), and ρ 2 any polymer from (y, t 2 ) to (z, t 3 ).The union of these two subsets of R 2 is an n-path from (x, t 1 ) to (z, t 3 ).We call this n-path the concatenation of ρ 1 and ρ 2 and denote it by ρ 1 • ρ 2 .The weight of ρ 1 • ρ 2 is W (y,t 2 ) n;(x,t 1 ) + W (z,t 3 ) n;(y,t 2 ) .This additivity is the reason that the endpoint v was excluded from the definition of path energy in Section 1.1.

3.2.
Polymer ordering lemmas.The first lemma roughly says that if two polymers intersect at two points during their lifetimes, then they are identical between these points.
To simplify notation in the proof, we write ρ 1 = ρ ←;(y 1 ,s 1 ) n;(x 1 ,t 1 ) and ρ 2 = ρ ←,(y 2 ,s 2 ) n;(x 2 ,t 2 ) .Proof of Lemma 3.1.First, for any polymer ρ, call a point u ∈ ρ a Poisson point of ρ if T −1 n (u) ∈ Π ∩ Γ, where Γ is the geodesic T −1 n (ρ) and Π is the underlying unit rate Poisson point process.Also, for r 1 , r 2 ∈ ρ, let ρ[r 1 , r 2 ] denote the part of the polymer between the points r 1 and r 2 , and let #ρ[r 1 , r 2 ] denote the number of Poisson points that lie in ρ where z 1 and z 2 appear in the lemma's statement.For, if not, without loss of generality assume that #ρ 1 [z 1 , z 2 ] < #ρ 2 [z 1 , z 2 | and let u 1 and v 1 be the Poisson . This illustrates Lemma 3.2.The points of the underlying Poisson process lying on a polymer are marked by dots, and the polymer is obtained by linearly interpolating between the points.The figure shows that both the paths cannot be leftmost polymers between their respective endpoints, since by joining the dashed lines, one obtains an alternative increasing path where the Poisson points between the intersecting points z 1 and z 2 in the two polymers are interchanged.
points of ρ 1 immediately before z 1 and immediately after z 2 ; and let u 2 and v 2 be the Poisson points of ρ 2 immediately after z 1 and immediately before z 2 : see Figure 2. Then joining u 1 to u 2 and v 1 to v 2 (shown in the figure by dashed lines), one gets an alternative path ρ between (x 1 , t 1 ) and (y 1 , s 1 ) that has more Poisson points than ρ 1 , thereby contradicting that ρ 1 is a polymer between (x 1 , t 1 ) and (y 1 , s 1 ).Thus, #ρ 1 [z 1 , z 2 ] = #ρ 2 [z 1 , z 2 |.Since both ρ 1 and ρ 2 are leftmost polymers between their respective endpoints, we see that . This proves the lemma.The next result roughly says that two polymers that begin and end at the same heights, with the endpoints of one to the right of the other's, cannot cross during their shared lifetime.Lemma 3.2 (Polymer Ordering).Fix n ∈ N. Consider points (x 1 , t 1 ), (x 2 , t 1 ), (y 1 , t 2 ), (y . Proof of Lemma 3.2.Supposing otherwise, there exists z = (x, y) ∈ ρ 2 such that x < ρ 1 (y).But then there exist z 1 , z 2 ∈ ρ 1 ∩ ρ 2 straddling the point z.By Lemma 3.1, By ordering, a polymer whose endpoints are straddled between those of a pair of polymers becomes sandwiched between those polymers.Corollary 3.3.Fix n ∈ N. Consider points (x 1 , t 1 ), (x 2 , t 1 ), (x 3 , t 1 ), (y 1 , t 2 ), (y 2 , t 2 ), (y The same result holds for rightmost polymers. Proof.By Lemma 3.2, ρ 1 (t) ≤ ρ 2 (t) ≤ ρ 3 (t) .The result now follows immediately.In this section, we show that the sequence ρ ← n : n ∈ N of leftmost n-polymers from (0, 0) to (0, 1) is tight, and any weak limit is Hölder 2/3−-continuous with a polylogarithmic correction of order 1/3.The main two ingredients in this proof are the local regularity estimate Theorem 2.5 and the polymer ordering Lemma 3.2.First, we bound the fluctuation of the polymer near any given point z ∈ [0, 1].Proposition 4.1.There exist positive constants n 0 , s 1 and c such that, for all n ≥ n 0 , s ≥ s 1 , z ∈ [0, 1] and 0 ≤ t ≤ 1 − z, The same statement holds for ρ → n .
As we now explain, the proposition will be proved by reducing to the case that z = 0, when the result follows from Theorem 2.5.For any fixed z ∈ (0, 1), Theorem 2.5 again guarantees that the polymer ρ ← n is at distance at most s from the point (0, z) with probability at least 1 − e −cs 3 .We break the horizontal line segment of length 2s centred at (0, z) into a sequence of consecutive intervals of length 2 −1 st 2/3 , and consider the leftmost polymers starting from each of these endpoints and ending at (0, 1), as in Figure 3. Due to the Corollary 3.3 of the polymer ordering Lemma 3.2, a big fluctuation of ρ ← n between times z and z + t creates a big fluctuation for one of the polymers starting from these deterministic endpoints.The probability of the latter fluctuations is controlled via Theorem 2.5 and since the number of these polymers is of the order of t −2/3 , a union bound gives (25).
Proof of Proposition 4.1.First observe that for s > (nt) 1/3 , the probability in ( 25) is zero by the definition of the scaling map T n in (2) and the geodesics being increasing paths.Hence we assume that s ≤ (nt) where TF (0,1) n;(0,0) is defined in (5).Hence, applying Theorem 2.6 with the parameter specifications t = 1 and k = 2 −1 8 −2 s, we get that (25) holds for all n, s large enough.Hence we assume that t ≤ 8 −3 .Also, let us assume for now that z ∈ [0, 2 −1 ].
The proof of Proposition 4.1 is illustrated here.We mark the line segment L with a number of equally spaced points.As the leftmost polymer from (0, 0) to (0, 1) passes between two such points on the line L, it is, in view of polymer ordering, sandwiched between the two leftmost polymers, shown as dotted lines, originating from those points and ending at (0, 1).Hence it is sufficient to bound the fluctuations of the polymers originating from these equally spaced points on L.
By Corollary 3.3, on E, Also, for any fixed i ∈ 0, 4t −2/3 , let (i) = (0,1) (x i ,z) be the straight line segment joining (x i , z) and (0, 1).Then, since z ∈ [0, 2 −1 ] and t ≤ 8 −3 , for any Thus, on the event E, by (26), From here, it follows by taking a union bound that for some absolute positive constant c and all n ≥ 2n 0 .Here the last inequality follows by applying Theorem 2.5 to each of the polymers ρ (i) .For given i, set the parameters n = n, Thus one can apply Theorem 2.5 to get the above inequality for all nt 1,2 , and follow the above argument.
Next we show the tightness of the members of the sequence {ρ ← n } n∈N as elements in the space (C[0, 1], • ∞ ).We prove that Proposition 4.1 guarantees that Kolmogorov-Chentsov's tightness criterion is satisfied.Proof of Theorem 1.1(a).Fix n ≥ n 0 and any λ > 0. Fix t ∈ (0, 1] small enough that λt −2/3 ≥ s 1 , where n 0 and s 1 are as in Proposition 4.1.Also fix some M ∈ N large enough that 2M − 2/3 > 1.Then it follows from Proposition 4.1 that for any z, z 4.1.Modulus of continuity.Here we prove Theorem 1.1(b), thus finding the modulus of continuity for any weak limit of a weakly converging subsequence of {ρ ← n } n∈N .We will follow the arguments used to derive the Kolmogorov continuity criterion, where one infers Hölder continuity of a stochastic process from moment bounds on the difference of the process between pairs of times.Thus we introduce the set of dyadic rationals Next is the first step towards proving the modulus of continuity.Lemma 4.2.Let ρ ← * be the weak limit of a weakly converging subsequence of {ρ ← n } n∈N .Then there exists a universal positive constant C (not depending on the particular weak limit ρ ← * ) such that, almost surely, for some random m 0 (ω) ∈ N and for all s, t Proof.For m ∈ N, let S m be the set of all intervals of the form [j2 −m , (j + 1) is an open set.Thus, by the Portmanteau theorem, Now, for all m large enough that (log 2 m ) 1/3 ≥ s 1 , where s 1 is as in Proposition 4.1, and all n ≥ n 0 , applying Proposition 4.1 and a union bound, .
Proof of Theorem 1.1(b).For any s, t ∈ [0, 1] satisfying s < t and |s − t| ≤ 2 −m 0 (ω) , choose , the theorem follows by taking the limit as k → ∞.The same argument applies without any change for the rightmost polymers as well.
5. Exponent pair (2/3, 1/3) for maximum fluctuation over short polymers: Proof of Theorem 1.2 In this section, we shall prove Theorem 1.2.It is the upper bound that is the more subtle.Recall the notation of transversal fluctuations from ( 4) and ( 5), AdEndPair n (t) from ( 6) and MTF n (t) from (7).
Here is the idea behind the proof.Proposition 1.4 offers a lower bound on the transversal fluctuation of a polymer between two given points.By considering order-t −1 endpoint pairs with disjoint intervening lifetimes of length t, we obtain a collection of independent opportunities for the fluctuation lower bound to occur.By tuning the probability of the individual event to have order t, at least one among the constituent events typically does occur, and the lower bound in Theorem 1.2 follows.
On the other hand, suppose that a big swing in the unit order region happens between a certain endpoint pair, with an intervening duration, or height difference, of order t.Members of the endpoint pair may be exceptional locations when viewed as functions of the underlying Poisson point field, both in horizontal and vertical coordinate.Thus, the upper bound in Theorem 1.2 does not follow directly from a union bound of a given endpoint estimate over elements in a discrete mesh, since such a mesh may not capture the exceptional endpoints.However, polymer ordering forces exceptional behaviour to become typical and to occur between an endpoint pair in a discrete mesh.To see this, assume that the original polymer between exceptional endpoints makes a big left swing.(Figure 4 illustrates the argument.)We take a discrete mesh endpoint pair whose lifetime includes that of the original polymer but has the same order t, and whose lower and upper points lie to the left of the original endpoint locations, about halfway between these and the leftmost coordinate visited by the original polymer.Then we consider the leftmost mesh polymer at the beginning and ending times of the original polymer.If the mesh polymer is to the right of the original polymer at any of these endpoints, then the mesh polymer has already made a big rightward swing at one of these endpoints.If, on the other hand, the mesh polymer is to the left of the original polymer at both the endpoints of the original polymer, then by polymer ordering Lemma 3.2, the mesh polymer cannot cross the original polymer during the latter's lifetime.Hence the big left swing of the original polymer forces a significant left swing for the mesh polymer as well.
Proof of Theorem 1.2.The lower bound follows in a straightforward way from Proposition 1.4.For any t ∈ (0, 1) and i ∈ 0, 1, 2, • • • , t −1 − 1 , define F i,t,n = TF (0,(i+1)t n;(0,it) ≥ ct 2/3 log t −1 1/3 .For given such (t, i), we apply Proposition 1.4 with parameter settings n = n, t 1 = it, t 2 = (i + 1)t and s = c(log t −1 1/3 , to find that, when c(log t −1 ) 1/3 ≥ s 0 and n ≥ max{α −3 0 c 3 t −1 log t −1 , n 0 t −1 }, , where the proposition specifies the quantities α 0 , n 0 and s 0 . i,j f (1)   i,j i,j and f i,j (shown in blue) is to the left of u and v at t 1 and t 2 respectively, then the blue polymer stays to the left of the red polymer between times t 1 and t 2 by polymer ordering.Thus the big left fluctuation transmits from the red to the blue polymer.If, however, the blue polymer reaches to the right of either u or v, then it creates a big right fluctuation for the blue polymer.Thus by bounding the fluctuations of a small number of polymers between deterministic endpoints, one can bound the fluctuation between all admissible endpoint pairs.Thus, for all t ≤ e −(c −1 s 0 ) 3 and i By choosing c > 0 small enough that c * c 3 < 1, one has lim inf n t −1 P(F 0,t,n ) → ∞ as t 0. For such c > 0, using the definition (7) of MTF n (t) and independence of the events Thus, lim sup ≤ lim sup n exp − t −1 P(F 0,n ) → 0 , the latter convergence as t 0.
Now we show the upper bound.Fix t ∈ (0, 1] small enough that ψt ≤ t 2/3 , where the parameter ψ appears in the definition (6) of AdEndPair n (t).
Then we claim that, whatever the value of C > 0, To see (30), define the vertical lines: Then, on the event B i,j , there exists a pair of points (u, t 1 ), (v, t 2 ) ∈ A i,j such that either ρ n;(u,t 1 ) intersects L 2 .We now show that, when ρ ←;(v,t 2 ) n;(u,t 1 ) intersects L 2 , the event D (1) i,j occurs.Let i,j be the line segment joining e (1) i,j and f n;(u,t 1 ) intersects L 2 , the event D (2) i,j occurs.We have proved (30).
For any compatible pair of points (u, v) ∈ AdEndPair n (t), there exists a pair (i, j) for which u, v ∈ A i,j ; here we use ψt ≤ t 2/3 .Hence, where (30) was used in the latter inclusion.Thus, with c, k 0 , n 0 as in the statement of Theorem 2.6, for any fixed t small enough that log t −1 ≥ 2 2 k 3 0 , and all n ≥ n 0 (2t) −1 , we have by a union bound and the translation invariance of the environment, Here the second inequality follows from Theorem 2.6 with t = 2t, k = 2 −2/3 (2 −1 C − 1) log t −1 1/3 and n = n being the parameter settings.The assumptions log t −1 ≥ 2 2 k 3 0 , and n ≥ n 0 (2t) −1 ensure that n ≥ n 0 t −1 and k ≥ k 0 for any C ≥ 2.
Finally, choosing C large enough that c (C/2 − 1) 3 > 5/3, we learn that This completes the proof of Theorem 1.2.

6.
Exponent pair (1/3, 2/3) for polymer weight: Proof of Theorem 1.3 A lemma and two propositions will lead to the proof of Theorem 1.3 on the Hölder continuity of [1, 2] → R : t → Wgt n (t), the polymer weight profile under vertical displacement.Lemma 6.1.There exist positive constants n 0 , r 0 , s 0 , c 0 such that, for all n ≥ n 0 We postpone the proof to Section 6.1 and first see how the lemma implies the upper bound in Theorem 1.3.This bound follows from Lemma 6.1 similarly to how Theorem 1.1 is derived from Proposition 4.1.Proposition 6.2.The sequence {Wgt n } n∈N is tight in (C[1, 2], • ∞ ).Moreover, if Wgt * is the weak limit of a weakly converging subsequence of {Wgt n } n∈N , then there exists a positive constant C not depending on the particular weak limit Wgt * such that, almost surely, Lemma 6.1 holds only for t ∈ [max{r 0 n −1 , 10 −3/2 s 3/2 n −1 }, 2 − z] for some fixed constant r 0 > 0, and not for all t ∈ [0, 1 − z], as was the case in Proposition 4.1.Hence, we directly show tightness in the following proof instead of applying Kolmogorov-Chentsov's tightness criterion.
From here, using (37) and that our given choice of the constant c ensures 2 3/2 c 2 c 3/2 < 1, we get As the second term decays much faster than the first, choosing M 0 large enough so that the second term is smaller that 2 −1 cm −1/3 (log m) 2/3 gives the result.
Proof of Theorem 1.3.This result follows from Proposition 6.2 and Proposition 6.3.
The latter two probabilities will each be bounded above by a union bound over several applications of Theorem 2.3.Addressing the first of these probabilities to begin with, we set parameters for The key box for the proof is Strip, now specified to be [−s, s] × [0, 1].Proposition 1.4 is, after all, a lower bound on the probability that there exists a polymer between (0, 0) and (0, 1) that escapes Strip.
We divide Strip into three further boxes, writing Mid for the box [−s, s] × [1/3, 2/3], and South and North for the boxes obtained from Mid by vertical translations of −1/3 and 1/3.We further set West to be the box obtained from Mid by a horizontal translation of −2s.See Figure 6.
Recall that, when (x, t 1 ) and (y, t 2 ) verify n 1/3 t 1,2 ≥ |y − x|, we denote the polymer weight with this pair of endpoints by W (y,t 2 ) n;(x,t 1 ) .We now use a set theoretic notational convention to refer in similar terms to the set of weights of polymers between two collections of endpoint locations.Indeed, let I and J be compact real intervals.We will write W (J,t 2 ) n;(I,t 1 ) = W (y,t 2 ) n;(x,t 1 ) : x ∈ I, y ∈ J ; we will ensure that whenever this notation is used, (x, t 1 ) n (y, t 2 ) for all x ∈ I and y ∈ J in the sense of Subsection 1.1.2.When an interval is a singleton, I = {x} say, we write (x, t 1 ) instead of ({x}, t 1 ) when using this notation.
To any box B and s ∈ R, we define the event High(B, s) that the weight of some path that is contained in B with starting point in the lower side of B and ending point in the upper side of B is at least s.

Figure 1 .
Figure 1.The scaling map T n applied to the left figure produces the figure on the right.The point e in the geodesic Γ is the preimage of the point (ρ(t), t) in the polymer ρ.

Figure 4 .
Figure 4.The figure illustrates the proof of the upper bound in Theorem 1.2.If the leftmost polymer between (u, t 1 ) and (v, t 2 ) (shown in red) makes a huge leftward fluctuation and the leftmost polymer between points e

Figure 6 .
Figure 6.In Case High, the high weight path ρ is extended to form a path from (0, 0) to (0, 1) whose weight exceeds that of any path between these points that remains in Strip = North ∪ Mid ∪ South.
Poissonian last passage percolation.Moreover, the refined bounds of Theorem 2.3 give corresponding improvements for Poissonian LPP: see [2, Remark 1.5 This local fluctuation estimate for the leftmost geodesic in (20) was proved for exponential directed last passage percolation in [2, Theorem 2 and Corollary 2.4].The proof goes through verbatim for the leftmost (and also the rightmost) geodesic in ) (t) for all t ∈ [t 1 , t 2 ].Thus ρ intersects L 2 as well, and hence D