Upper tail large deviations in Brownian directed percolation

This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.


Introduction and results
Brownian directed percolation is a model of directed last passage percolation in a white noise environment on RˆZ`. Given a family of i.i.d. standard two-sided Brownian motions tB i u 8 i"0 , we take the convention that B k ps, tq " B k ptq´B k psq and define last passage times from pu, kq to pt, nq, with u ď t P R and k ď n P Z`, by L k,n pu, tq :" sup u"s k´1 ăs k ă¨¨¨ăs n´1 ăsn"t $ & % n ÿ j"k B j ps j´1 , s j q , . -.
We will distinguish the last passage times from p0, 1q to pt, nq with the notation L n ptq : L n ptq :" max 0ăs 1 ă¨¨¨ăs n´1 ăt tB 1 p0, s 1 q`¨¨¨`B n ps n´1 , tqu . (1.1) The model was originally introduced in the queueing literature in [10]. It is rich with connections to other families of models including directed polymers and last passage percolation [1,5,14,18], queueing theory [10,18] and random matrix theory [2,11,19]. The random matrix connection is of particular interest: a distributional equivalence between the last passage time L n ptq and the largest eigenvalue of an appropriate GUE matrix was discovered independently by Baryshnikov [2, Theorem 0.7] and Gravner, Tracy, and Widom [11] and soon after extended by O'Connell and Yor [19].
Given how well understood the distribution of the largest eigenvalue of a GUE matrix is, this identity yields an enormous amount of information about the directed percolation model. For example, the analogue of the free energy limit (in distribution and hence in probability) for this model follows from classical results in random matrix theory. The almost sure version of this limit for L n p¨q is due to Hambly, Martin, and O'Connell [12], Theorem 8]: almost surely, for all t ą 0, Through the distributional equivalence, fluctuations around this limit are given by the Tracy-Widom GUE distribution [24].
Large deviation results are known at both rates n and n 2 . These results depend on the large deviation principle for the empirical distribution of a random matrix drawn from the Gaussian Unitary Ensemble, which is due to Ben Arous and Guionnet [4,Theorem 1.3]. The (rate n) upper tail large deviation rate function can then be derived as in the computation of the corresponding rate function for the largest eigenvalue of a random matrix drawn from the Gaussian Orthogonal Ensemble in [3,Theorem 6.2]. The precise expression here can be found in the lecture notes of Ledoux on deviation inequalities for largest eigenvalues [17, (1.25)]. Theorem 1.1. For any r ě 0, lim nÑ8´n´1 log P`n´1L n pnq ě 2p1`rq˘" 4 ż r 0 a xpx`2qdx :" J GU E prq.
In this paper, we provide a simple alternative proof of Theorem 1.1 using ideas which have previously been used to derive large deviation principles for the free energy of certain solvable positive and zero temperature directed polymer models in [6,8,9,15,22,23]. The proof presented below shows that the limit in the statement of the theorem exists by subadditivity argument, from which we immediately derive the following corollary, which is also observed in [17, (2.6)]: Corollary 1.2. For any n and r ě 0, The key input needed for the approach taken in this paper is an analogue of Burke's theorem from queueing theory, which allows us to define a version of the percolation model in which appropriate differences of passage times are stationary.
These have the interpretation as the 'departures' and 'queue length' processes for the server at station k in a stationary queueing Brownian queueing model. These definitions follow naturally from applying Donsker's theorem to the corresponding definitions in the classical M/M/1 queue. As is shown for example in [18, Theorem 2, Section 4], (see also [13] for a more general result) applying Donsker's theorem to Burke's theorem for the M/M/1 queue leads to the following Brownian analogue of Burke's theorem: For each t ě 0, the family tq µ k ptqu 8 k"1 consists of i.i.d. Exponential random variables with mean µ´1.
With these definitions, an induction argument shows that In particular, ř n k"1 q µ k p0q " L µ n p0q. We think of paths in this extended directed percolation model as being indexed by the points where they exit the lines t0, . . . , nu. By grouping paths into those that exit line 0 before time 0 and those that exit after, we obtain The decomposition in (1.4) can be viewed as describing a 'stationary' point-to-point polymer on R`ˆZ`with i.i.d. exponential boundary conditions tL µ n`1 p0q´L µ n p0qu nPN on the vertical axis and drifted Brownian boundary conditions tµt´B 0 ptqu tě0 on the horizontal axis. The model is stationary in the sense that tL µ n`1 ptq´L µ n ptqu nPZ`" tq µ n`1 ptq : n P Z`u is an i.i.d. Exponential family for each t ą 0. We will combine the queueing picture with this finite n variational problem in order to obtain a variational problem for the Lyapunov exponents in this model which will allow us to prove Theorem 1.1. It is convenient to write the previous expression in a way that separates the terms ř n k"1 q µ k ptq and B 0 ptq´µt, which are not independent: The key point in this decomposition is that for each s 0 ą 0, the random variables B 0 ptq´B 0 ps 0 q and L 1,n ps 0 , tq are independent and for each j P t1, . . . , nu, the random variables B 0 ptq, ř j k"1 q µ k p0q, and L j,n p0, tq are mutually independent. This independence can be seen by recalling that the Brownian motions tB i u 8 i"0 are independent and observing that σpB i psq : s ď 0, i P Z`q and σpB i psq : s ě 0, i P Z`q are independent. This decomposition will lead to a variational problem which can be used to prove Theorem 1.1. Once we have proven Theorem 1.1, we can bootstrap that result and the decomposition in (1.4) to compute the corresponding positive moment Lyapunov exponents for the stationary model.
One of the main goals of this paper is to keep the proofs short and non-technical. For this reason, we stop at computing the Lyapunov exponents for the stationary model and do not prove the corresponding upper tail rate function limit. The essential technical difficulty is that we no longer have access to subadditivity in the stationary model, which would have given us a priori existence, finiteness, and convexity of the upper tail rate function. If we knew those properties, then Theorem 1.4 would give the rate function by taking a Legendre transform. Without this, the rate function can be computed following steps similar to the proof of Theorem 1.4, but working instead with lim sup and lim inf upper tail rate functions directly. The analysis and calculus needed to derive and solve the resulting variational problem become more involved than in the proof of Theorem 1.4. See [9,15] and in particular the proof of [9, Theorem 2.14] for a similar argument.  " e λL t ns u pntq ı , J s,t prq :" lim nÑ8´1 n log P`L t ns u pntq ě nrȇ xist and are finite. Moreover J s,t prq ě 0. For each λ ą 0, the map ps, tq Þ Ñ Λ s,t pλq for ps, tq P p0, 8q 2 is positively homogeneous of degree one, superadditive, concave, and continuous. For ps, t, rq P p0, 8q 2ˆR , the map ps, t, rq Þ Ñ J s,t prq is positively homogeneous of degree one, subadditive, convex, and continuous. For each ps, tq, J s,t prq " 0 for r ď 2 ? st and r Þ Ñ J s,t prq is non-decreasing.
Proof. Note that the pre-limit expression in the definition of J s,t prq is non-negative. For all of the conclusions except finiteness of Λ s,t pλq and the last two properties of J s,t prq, it then suffices to show that the maps e λL t s u ptq ı , ps, t, rq Þ Ñ´log P`L t s u ptq ě rȃ re superadditive on r1, 8qˆp0, 8q and subadditive on r1, 8qˆp0, 8qˆR respectively. See [16,Theorem 16.2.9] and the comment following the proof. Note that a subadditive function which is positively homogeneous of degree one is convex. Take s 1 , s 2 ě 1, t 1 , t 2 ą 0 and r 1 , r 2 P R. We have the inequality L tps 1`s2 q u pt 1`t2 q ě L t s 1 u pt 1 q`L t s 1 u,tps 1`s2 q u pt 1 , pt 1`t2 qq where the last two terms are independent. Using translation invariance, independence, and monotonicity of L n ptq in n, we have Finiteness of Λ s,t pλq for all λ ą 0 follows from L n ptq ď ř n i"0 2 max 0ďrďt |B i prq|. The properties of J s,t prq follow from the almost sure limit (1.2), continuity, and the fact that the pre-limit expression is non-decreasing in r.
The next result shows that the decomposition in (1.5) implies that Λ s,t pλq is the solution to an invertible variational problem. This type of decomposition and versions of the argument that follows are the key steps in the papers [6,8,9,15]. Lemma 2.3. For each s, t ą 0 and λ P p0, µq, Proof. We begin with the coupling (1.5). It follows that for any r P r0, tq, u P r0, sq and n large enough, The random variables B 0 pntq´B 0 pnrq and L 1,n pnr, ntq are independent because B 0 p¨q is independent of tB j p¨qu 8 j"1 . The random variables ř t nu u k"1 q µ k p0q, L t nu u,n p0, tq, and B 0 ptq are independent because ř t nu u k"1 q µ k p0q is measurable with respect to σpB j ptq : t ď 0, j P Z`q, L t nu u,n p0, tq is measurable with respect to σpB j ptq : t ě 0, j P Nq, and B 0 ptq is measurable with respect to σpB 0 ptq : t ą 0q. Taking logs, dividing by n and sending n Ñ 8, and optimizing over u and r, we immediately obtain ě in the statement of the theorem.
Let tr i u M i"1 and tu i u M i"1 be uniform partitions of r0, ts and r0, ss respectively. Notice that max 0ďrďt nµpr´tq`B 0 pntq´B 0 pnrq`L 1,t ns u pnr, ntq ( Similarly, we have , . -.

It follows from these inequalities and independence that
" e λL t nu i´1 u,t ns u p0,ntq ı By the reflection principle and the assumption that r i´ri´1 " t M , we have Take logs, divide by n and send n Ñ 8 to obtain Sending M Ñ 8 completes the proof.
Variational problems of the type in Lemma 2.3 appear for the Lyapunov exponents and free energies (resp. time constants) of directed polymers (resp. percolation models) which have associated stationary models that satisfy appropriate analogues of the Burke property. Up to a change of variables, a deformation of the region on which the maximization takes place, and homogeneity of Λ s,t pλq in ps, tq, this variational expression gives a Legendre-Fenchel duality between directions ps, tq and values of µ ą λ. See for example [7,Section 5] for this point of view. Alternatively, this variational problem can be solved directly with some easy calculus. This is done in some generality in [15,Proposition 3.10], so we appeal to that result here.
Corollary 2.4. For any s, t, λ ą 0, Proof. The first equality follows from Lemma 2.3 and [15, Proposition 3.10] with I " tµ ą λu, hpµq "´λ Proof. The result for λ ď 0 follows from the observations J s,t prq ě 0 for all r, J s,t prq " 0 for r ď 2 ? st and r Þ Ñ J s,t prq is non-decreasing. Take λ, K ą 0, and let tm i u M i"1 be a uniform partition of r0, Ks. The exponential Markov inequality yields for each r ą 0 λr´J s,t prq ď Λ s,t pλq. Take logs, divide by n and send n Ñ 8 to obtain Equation ( Having computed Λ s,t pλq, (1.4) now leads to a variational problem for the Lyapunov exponents in the stationary model for each µ ą λ. Using Corollary 2.4 we may extend Λ s,t pλq continuously to Λ 0,t pλq " λ 2 t 2 and Λ s,0 pλq " 0.
Lemma 3.1. For each µ, s, t ą 0 and λ P p0, µq, Proof. The proof of the first equality is essentially the same as in the proof of Lemma 2.3, except that one must work with lim inf and lim sup. For example, for any r P r0, tq, u P r0, sq, and n sufficiently large, we have Take logs, divide by n, take lim inf, and optimize to obtain We omit the reverse inequality which similarly follows from the type of arguments in the proof of Lemma 2.3. For the second equality, it is convenient to substitute r Þ Ñ t´r and u Þ Ñ s´u. The second equality comes from dividing the minimum into the regions z ď µ and z ą µ. A similar argument using the same variational expression and dividing into z ď µ´λ and z ą µ´λ shows that max 0ďuďs " u log µ µ´λ`Λ s´u,t pλq * " " tˆ´λ 2 2`µ λ˙`s log µ µ´λ * min zďµ´λ " tˆλ 2 2`z λ˙`s log z`λ z * .
In both of these expressions, the first term is feasible in the minimum over z by taking z " µ or z " µ´λ. To complete the proof, note that the function being minimized in the second variational expression Λ s,r pλq in Corollary 2.4 is strictly convex and minimizers exist.
We complete this section and the paper by addressing the exponents λ ě µ. This is an immediate corollary of the previous lemma. Sending λ Ò µ gives the result.