1 Introduction

This paper studies the model of a 1 + 1 dimensional semi-discrete directed polymer in a random environment due to O’Connell and Yor [25]. Directed polymers in random environments were introduced in the statistical physics literature in [17], with mathematical work following in [5, 18]. A physically motivated mathematical introduction to this family of models can be found in the survey article [9]. These models have attracted substantial attention recently, in part due to the conjecture that under suitable regularity assumptions they lie in the KPZ universality class. For a discussion of this conjecture and further references, we refer the reader to [10].

Our study will focus on the point-to-point polymer partition function of this model, which [25] defines for \(n \in \mathbb {N}\) and \(\beta > 0\) as

$$\begin{aligned} Z_n(\beta )&= \int _{0 < s_1 < \cdots < s_{n-1} < n}\exp \left[ \beta \left( B_1(0,s_1) + \cdots + B_n(s_{n-1},n) \right) \right] ds_1 \ldots ds_{n-1}, \end{aligned}$$
(1)

where \(\{B_i\}_{i=1}^\infty \) is a family of i.i.d. standard Brownian motions. This partition function is the normalizing constant for a quenched polymer measure on non-decreasing càdlàg paths \(f:\mathbb {R}_+ \rightarrow \mathbb {N}\) with \(f(0) = 1\) and \(f(n) = n\). Up to a constant factor, \(Z_n(\beta )\) is the conditional expectation of a functional of a Poisson path on the event that the path is at \(n\) at time \(n\). More precisely, let \(\pi (\cdot )\) be a unit rate Poisson process on \(\mathbb {R}_+\) which is independent of the family \(\{B_i\}_{i=1}^\infty \) and denote by \(\mathcal {E}\) the expectation with respect to the law of this Poisson process. With the notation

$$\begin{aligned} A_{n,t} = \{s_1, \ldots s_{n-1} : 0 < s_1 < \cdots < s_{n-1} < t \}, \end{aligned}$$
(2)

we have

$$\begin{aligned} \mathcal {E}_{\pi (0)=1} \left[ e^{\int _0^n \beta dB_{\pi (s)}(s)} | \pi (n) =n \right]&= |A_{n,n}|^{-1} Z_n(\beta ). \end{aligned}$$
(3)

The prefactor of \(|A_{n,n}|^{-1}\) accounts for the fact that the ordered jump points of \(\pi \) on \([0,n]\) conditioned on \(\pi (n) = n\) are uniformly distributed on the Weyl chamber \(A_{n,n}\).

The O’Connell–Yor polymer model was originally introduced in [25] in connection with a generalization of the Brownian queueing model. Based on the work of Matsumoto and Yor [21], O’Connell and Yor were able to show the existence of a stationary version of this model satisfying an analogue of Burke’s theorem for M/M/1 queues. The Burke property makes the O’Connell–Yor polymer one of the four polymer models considered exactly solvable, the others being the continuum directed polymer studied in [1], the log-gamma polymer introduced by Seppäläinen in [28], and the strict-weak gamma polymer studied in [11] and [24]. A precise statement of the Burke property for this model and an outline of how this property leads to the main result of this paper is given in Sect. 2.2. Subsequent work on the representation theoretic underpinnings of the exact solvability of these models can be found in the work of Borodin and Corwin on Macdonald processes [6] and the work of O’Connell connecting this model to the quantum Toda lattice [23].

As one of the few tractable models in the KPZ universality class, this polymer model has been extensively studied: Moriarty and O’Connell [22] rigorously computed the free energy; Seppäläinen and Valkó [29] identified the scaling exponents; Borodin, Corwin, and Ferrari [8] showed that the model lies in the KPZ universality class by proving the Tracy–Widom limit for the free energy fluctuations; and Borodin and Corwin [7] proved a contour integral representation for the integer moments and computed their large \(n\) asymptotics.

The main results of this paper are Theorems 2.2 and 2.3, which compute the moment Lyapunov exponents and large deviation rate function with normalization \(n\) for the free energy respectively. Theorem 2.2 can be thought of as an extension of the asymptotics studied in [7]. There, the authors use a contour integral representation for the integer moments of \(Z_n(\beta )\) to compute the limit

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n} \log E\left[ Z_{n}(\beta )^k\right] \end{aligned}$$

for any \(k \in \mathbb {N}\). In [7], Appendix A.1], as part of a replica computation of the free energy, they conjecture an analytic continuation of their formula to \(k >0\). We are able to compute the above limit for all \(k \in \mathbb {R}\), confirming this conjecture.

The proofs of Theorems 2.2 and 2.3 follow an approach introduced by Seppäläinen in [27]. Georgiou and Seppäläinen used this method to compute the large deviation rate function with normalization \(n\) for the free energy in the log-gamma polymer in [14]. The key technical condition making this scheme tractable is the independence provided by the Burke property, which the log-gamma polymer shares with the O’Connell–Yor polymer. It is therefore natural to expect that the techniques of [14] should also apply in this setting; we take this as our starting point.

Physically, we can view the parameter \(\beta \) as an inverse temperature. In this sense, we can think of directed polymer models as positive temperature analogues of directed percolation. This leads to a natural coupling between the directed polymer and directed percolation models, which we take advantage of throughout the paper. The directed percolation model associated to the O’Connell–Yor polymer is Brownian directed percolation; see [16, 25] for a discussion of this model. A distributional equivalence between the last passage time in Brownian directed percolation and the largest eigenvalue of a standard GUE matrix was discovered independently by Baryshnikov [2], Theorem 0.7] and Gravner, Tracy, and Widom [15], both in 2001. The known large deviations [20] for top eigenvalues of GUE matrices give useful estimates in several of the proofs that follow. The precise results we use are collected in the ‘Bounds from the GUE connection’ section of the appendix.

This polymer and the log-gamma polymer are the only positive temperature polymer models for which precise large deviations have been studied. Precise left tail large deviations, which have non-universal scalings [3], remain open for both models. The lattice Gaussian directed percolation and polymer models in one spatial dimension have left tail large deviations with normalization \(\frac{n^2}{\log (n)}\) [12], Theorem 1.2] [3], p. 774], while the Brownian directed percolation model has left tail large deviations with normalization \(n^2\) [20], (1.26)]. It is natural to expect that the left tail large deviations for this model should follow those of Brownian directed percolation rather than the Gaussian lattice models, but we do not currently have a proof of this result.

Notation When \(b(\cdot )\) is Brownian motion and \(s \le t\), we adopt the convention \(b(s,t) = b(t) - b(s)\). For \(a,b \in [-\infty ,\infty ]\), we set \(a \vee b = \max (a,b)\) and \(a \wedge b = \min (a,b)\). The polygamma functions are denoted by \(\Psi _k(x)\), where \(\Psi _0(x) = \frac{d}{dx}\log \Gamma (x)\) and \(\Psi _n(x) = \frac{d}{dx} \Psi _{n-1}(x)\).

For \(f,g:\mathbb {R} \rightarrow (-\infty ,\infty ]\), the Legendre–Fenchel transform is defined by \(f^*(\xi ) = \sup _{x \in \mathbb {R}}\{x \xi - f(x)\}\) and the infimal convolution is defined by \(f \square g(x) = \inf _{y \in \mathbb {R}}\{f(x-y) + g(y)\}\). For properties of these operators, we refer the reader to [26].

A random variable has a \(\Gamma (\theta ,1)\) distribution if it has density \(\Gamma (\theta )^{-1}x^{\theta - 1}e^{-x}1_{\{x>0\}}\) with respect to the Lebesgue measure on \(\mathbb {R}\).

2 Preliminaries and Statement of Results

2.1 Definition of the Polymer Model and Statement of Results

Let \(\{B_i\}_{i=0}^\infty \) be a family of independent two-sided standard Brownian motions. Define partition functions for \(j,n \in \mathbb {Z}_+\) with \(j < n\) and \(s,t \in (0,\infty )\) with \(s < t\) by

$$\begin{aligned} Z_{j,n}(u,t)&= \int \limits _{u < u_j < \cdots < u_{n-1} < t} e^{B_j(u,u_j) + \sum _{i= j+1}^{n-1} B_i (u_{i-1}, u_i) + B_{n}(u_{n-1},t)}du_j \ldots du_{n-1}. \end{aligned}$$
(4)

For the case \(j = n\), we define

$$\begin{aligned} Z_{j,j}(u,t)&= e^{B_j(u,t)}. \end{aligned}$$
(5)

We will refer to the \(j,n\) variables as space and the \(u,t\) variables as time. Translation invariance of Brownian motion and our assumption that the environment is i.i.d. immediately imply that the distribution of the partition function is shift invariant.

It follows from Brownian scaling that for \(\beta > 0\) and \(n > 1\) we have

$$\begin{aligned} Z_n(\beta ) \mathop {=}\limits ^{\tiny {d}} \beta ^{-2(n-1)}Z_{1,n}(0,\beta ^2 n). \end{aligned}$$

For the remainder of the paper, we will only consider partition functions of the form \(Z_{j,n}(u,t)\); results for these partition functions can be translated into results for \(Z_n(\beta )\) using this distributional identity.

Next, we argue that the partition function is supermultiplicative: that is, for \(j, n,m \in \mathbb {Z}_+\), \(v \ge 0\), and \(u,t > 0\) ,

$$\begin{aligned} Z_{j,j+n+m}(v,v+t+u)&\ge Z_{j,j+n}(v,v+t)Z_{j+n,j+n+m}(v+t,v+t+u). \end{aligned}$$
(6)

For notational convenience, we will consider the case \(j= v = 0\). For \(t,u > 0\), we have

$$\begin{aligned} Z_{0,0}(0,t+u) = e^{B_0(0,t+u)} = e^{B_0(0,t)} e^{B_0(t,t+u)} = Z_{0,0}(0,t)Z_{0,0}(t,t+u). \end{aligned}$$

For \(m,n \in \mathbb {N}\) and \(t,u > 0\), we have

$$\begin{aligned}&Z_{0,n+m}(0,t+u)\nonumber \\&\quad = \int \limits _{0 < u_0 < \cdots < u_{n+m-1} < t+u} e^{B_0(0,u_0) + \sum _{i= 1}^{n+m-1} B_i (u_{i-1}, u_i) + B_{n+m}(u_{n+m-1},t + u)}du_0 \ldots du_{n+m-1} \nonumber \\&\quad \ge \int \limits _{\mathop {u_{n-1} < t < u_n}\limits ^{0 < u_0 < \cdots < u_{n+m-1} < t+u}} e^{B_0(0,u_0) + \sum _{i= 1}^{n+m-1} B_i (u_{i-1}, u_i) + B_{n+m}(u_{n+m-1},t+u)}du_0 \ldots du_{n+m-1} \nonumber \\&\quad = Z_{0,n}(0,t)Z_{n,n+m}(t,t+u). \end{aligned}$$
(7)

When \(m < n\) and \(t,u > 0\), we decompose \(\log Z_{m,n}(u,t)\) as follows:

$$\begin{aligned} \log Z_{m,n}(u,t)&= B_{n}(t) - B_m(u) + \log C_{m, n}(u,t) \end{aligned}$$
(8)

where

$$\begin{aligned} C_{m,n}(u,t)&= \int _u^t \int _{u_{m}}^t \ldots \int _{u_{n-2}}^t e^{B_m(u_m) + \sum _{i=m+1}^{n-1} B_i (u_{i-1}, u_i) - B_{n}(u_{n-1})}du_{n-1} \ldots du_{m+1} du_m \end{aligned}$$

is strictly increasing in \(t\) and strictly decreasing in \(u\). It follows that

$$\begin{aligned} Z_{m,n}(0,t+u)&\ge Z_{m,n}(0,t)Z_{n,n}(t,t+u), \\ Z_{m,n}(0,t+u)&\ge Z_{m,m}(0,t)Z_{m,n}(t,t+u). \end{aligned}$$

The free energy for (1) was computed in [22]. We mention that, as in [14], Lemma 4.1], once one knows the existence and continuity of the free energy, a variational problem similar to the one we study for the rate function in this paper can be used to compute the value of the free energy. We have

Lemma 2.1

([22]) Fix \(s,t \in (0,\infty )\). Then the almost sure limit

$$\begin{aligned} \rho (s,t)&= \lim _{n \rightarrow \infty } \frac{1}{n} \log Z_{1, \lfloor ns \rfloor }(0,nt) \end{aligned}$$

exists and is given by

$$\begin{aligned} \rho (s,t)&= \min _{\theta > 0} \left\{ \theta t - s \Psi _0(\theta ) \right\} = t \Psi _1^{-1}\left( \frac{t}{s}\right) - s \Psi _0\left( \Psi _1^{-1}\left( \frac{t}{s}\right) \right) . \end{aligned}$$

The main result of this paper is a computation of the real moment Lyapunov exponents of the parabolic Anderson model associated to (4) and, through an application of the Gärtner-Ellis theorem, the large deviation rate function with normalization \(n\) for the free energy of the polymer. Specifically, we have

Theorem 2.2

Let \(s,t \in (0,\infty )\) and \(\xi \in \mathbb {R}\). Then

$$\begin{aligned} \Lambda _{s,t}(\xi )&= \lim _{n \rightarrow \infty } \frac{1}{n} \log E\left[ e^{\xi \log Z_{1,\lfloor ns \rfloor }(0,nt)}\right] \\&= {\left\{ \begin{array}{ll} \xi \rho (s,t) &{} \xi \le 0 \\ \displaystyle \min _{\mu > 0} \left\{ t \left( \frac{\xi ^2}{2} + \xi \mu \right) - s \log \frac{\Gamma (\mu + \xi )}{\Gamma (\mu )} \right\} &{} \xi >0 \end{array}\right. } \end{aligned}$$

and \(\Lambda _{s,t}(\xi )\) is a differentiable function of \(\xi \in \mathbb {R}\).

Theorem 2.3

Fix \(s,t \in (0,\infty )\). The distributions of \(n^{-1}\log Z_{1,\lfloor ns \rfloor }(0,nt)\) satisfy a large deviation principle with normalization \(n\) and convex good rate function

$$\begin{aligned} I_{s,t} (x)&= {\left\{ \begin{array}{ll} \infty &{} x < \rho (s,t) \\ \Lambda _{s,t}^*(x) &{} x \ge \rho (s,t) \end{array}\right. }. \end{aligned}$$

Remark 2.4

The function being minimized for \(\xi > 0\) in Theorem 2.2 is convex and coercive, so the minimum is attained. The details this fact are worked out in the proof of Corollary 3.11.

2.2 Definition of the Stationary Model and Proof Outline

Let \(B(t)\) be a two-sided Brownian motion independent of the family \(\{B_i\}_{i=0}^\infty \) and for \(\theta >0\), \(t \in \mathbb {R}\) and \(n \in \mathbb {Z}_+\) define point-to-point partition functions by

$$\begin{aligned} Z_n^\theta (t)&= \int \limits _{- \infty < u_0 < u_1 < \cdots < u_{n-1} < t}e^{ \theta u_0 - B(u_0) + B_1(u_0, u_1) + \cdots + B_{n}(u_{n-1},t)} du_0 \ldots du_{n-1}, \end{aligned}$$

with the convention that

$$\begin{aligned} Z_0^\theta (t)&= e^{\theta t - B(t)}. \end{aligned}$$

We can think of \(Z_n^\theta (t)\) as a modification of the polymer in the previous subsection where we add a spatial dimension and start in the infinite past. For \(s,t > 0\) and \(n\) sufficiently large that \(ns\ge 1\), we obtain a decomposition of \(Z_{\lfloor ns \rfloor }^\theta (nt)\) into terms that involve the partition functions we are studying by considering where paths leave the potential of the Brownan motion \(B\):

$$\begin{aligned} Z_{\lfloor n s \rfloor }^\theta (nt)&= \int _0^{nt} Z_0^\theta (u)Z_{1,\lfloor n s \rfloor }(u,nt)du + \sum _{j=1}^{\lfloor ns \rfloor } Z_j^\theta (0)Z_{j, \lfloor n s \rfloor }(0,nt). \end{aligned}$$
(9)

This expression also leads to the interpretation of \(Z_n^\theta (t)\) as a modification of the point-to-point partition function discussed in the previous subsection where we have added boundary conditions.

We will refer to this model as the stationary polymer, where the term stationary comes from the fact that it satisfies an analogue of Burke’s theorem for M/M/1 queues. This fact is one of the main contributions of [25] and we refer the reader to that paper for a more in depth discussion of the connections to queueing theory. We follow the notation of [29], which contains the version of the Burke property that will be used in this paper. Define \(Y_0^\theta (t) = B(t)\) and for \(k \ge 1\) recursively set

$$\begin{aligned} r_k^\theta (t)&= \log \int _{-\infty }^t e^{Y_{k-1}^\theta (u,t) - \theta (t-u) + B_k(u,t)} du,\nonumber \\ Y_k^\theta (t)&= Y_{k-1}^\theta (t) + r_k^\theta (0) - r_k^\theta (t),\\ X_k^\theta (t)&= B_k(t) + r_k^\theta (0) - r_k^\theta (t); \nonumber \end{aligned}$$
(10)

then we have

Lemma 2.5

[29], Theorem 3.3] Let \(n \in \mathbb {N}\) and \(0 \le s_n \le s_{n-1} \le \cdots \le s_1 < \infty \). Then over \(j\), the following random variables and processes are all mutually independent.

$$\begin{aligned}&r_j(s_j) \ \mathrm{and } \ \{X_j(s): s\le s_j\} \ \mathrm{for } \ 1\le j\le n, \quad \{Y_n(s): s\le s_n\},\\&\qquad \mathrm{and} \quad \{Y_j(s_{j+1},s): s_{j+1}\le s\le s_j\} \ \mathrm{for } \, 1\le j\le n-1. \end{aligned}$$

Furthermore, the \(X_j\) and \(Y_j\) processes are standard Brownian motions, and \(e^{-r_j(s_j)}\) is \(\Gamma (\theta ,1)\) distributed.

An induction argument shows that

$$\begin{aligned} \sum _{k=1}^n r_k^\theta (t)&= B(t) - \theta t + \log Z_n^\theta (t). \end{aligned}$$
(11)

As we will see shortly, expression (9) would lead to a variational formula for the right tail rate function we are looking for in terms of the right tail rate function of \(Z_{\lfloor ns \rfloor }^\theta (nt)\). This right tail rate function would be tractable using (11) if \(B(nt)\) were independent of \(\sum _{k=1}^{\lfloor ns \rfloor } r_k^\theta (nt)\); as this is not the case, it is convenient to rewrite (9) in a form that separates these two terms:

$$\begin{aligned} e^{\sum _{k=1}^{\lfloor ns \rfloor } r_k^\theta (nt)}&= n \int _0^t \frac{Z_0^\theta (nu)}{Z_0^\theta (nt)}Z_{1, \lfloor ns \rfloor }(nu,nt) du + \sum _{j=1}^{\lfloor ns \rfloor } \frac{Z_j^\theta (0)}{Z_0^\theta (nt)} Z_{j,\lfloor ns \rfloor }(0,nt) . \end{aligned}$$
(12)

We now briefly outline the proof of Theorem 2.2. In order to compute the positive moment Lyapunov exponents, we consider the dual problem of establishing right tail large deviations for the free energy. As is typically the case, existence and regularity of the right tail rate function follows from subadditivity arguments. It then follows from (12) that this right tail rate function solves a variational problem in terms of computable rate functions coming from the stationary model. Taking Legendre–Fenchel transforms brings us back to the study of moment Lyapunov exponents and gives the variational problem a linear structure which makes it tractable.

For non-positive exponents, we use crude estimates on the partition function to identify the limit. We are able to do this because the left tail large deviations for the free energy are are strictly subexponential while the moment Lyapunov exponents are only sensitive to exponential scale large deviation.

3 A Variational Problem for the Right Tail Rate Function

3.1 Definitions and Notation

The goal of this subsection is to introduce the right tail rate function for the free energy, which we will denote \(J_{s,t}(x)\), and the rate functions coming from the stationary model which appear in the variational expression for \(J_{s,t}(x)\). We will defer some of the proofs of technical results about the existence and regularity of these rate functions to Appendix 1. We begin by defining these functions and addressing existence.

Theorem 3.1

For all \(s\ge 0\), \(t > 0\) and \(r \in \mathbb {R}\), the limit

$$\begin{aligned} J_{s,t}(r)&= \lim _{n \rightarrow \infty } - \frac{1}{n} \log P \left( \log Z_{1, \lfloor ns \rfloor }(0, nt) \ge nr \right) \end{aligned}$$

exists and is \(\mathbb {R}_+\) valued. Moreover, \(J_{s,t}(r)\) is continuous, convex, subadditive, and positively homogeneous of degree one as a function of \((s, t,r) \in [0,\infty ) \times (0,\infty ) \times \mathbb {R}\). For fixed \(s\) and \(t\), \(J_{s,t}(r)\) is increasing in \(r\) and \(J_{s,t}(r) = 0\) if \(r \le \rho (s,t)\).

The proof of Theorem 3.1 can be found in the ‘Existence and Structure of the Right Tail Rate Function’ section of Appendix 1.

Next, we define the computable rate functions from the stationary model. By the Burke property for the stationary model, the first limit below can be computed as the right branch of a Cramér rate function. For \(s,t > 0\), we set

$$\begin{aligned} U_s^{\theta }(x)&= - \lim \frac{1}{n} \log P \left( \sum _{k=1}^{\lfloor ns \rfloor } r_k^\theta (0)\ge nx \right) \\&= {\left\{ \begin{array}{ll} 0 &{} \quad x \le - s \Psi _0(\theta ) \\ x(\theta - \Psi _0^{-1}(-\frac{x}{s})) + s \log \frac{\Gamma (\theta )}{\Gamma (\Psi _0^{-1}(-\frac{x}{s}))} &{}\quad x > - s\Psi _0(\theta ) \end{array}\right. }\!,\\ R_t^\theta (x)&\quad = - \lim \frac{1}{n} \log P \left( B(nt) - \theta n t \ge nx \right) = {\left\{ \begin{array}{ll} 0 &{}\quad x \le -\theta t \\ \frac{1}{2}\left( \frac{x+ \theta t}{\sqrt{t}}\right) ^2 &{} x > -\theta t \end{array}\right. }\!\!. \end{aligned}$$

We may continuously extend \(U_s^\theta (x)\) to \(s = 0\) by setting

$$\begin{aligned} U_0^\theta (x)&= {\left\{ \begin{array}{ll} 0 &{}\quad x \le 0 \\ x \theta &{} \quad x > 0 \end{array}\right. }. \end{aligned}$$

We record the Legendre–Fenchel transforms of these functions below:

$$\begin{aligned} (U_s^\theta )^*(\xi )&= {\left\{ \begin{array}{ll} \infty &{}\quad \xi < 0 \text { or } \xi \ge \theta \\ s \log \frac{\Gamma (\theta - \xi )}{\Gamma (\theta )} &{}\quad 0 \le \xi < \theta \end{array}\right. }\!, \qquad (R_t^\theta )^*(\xi ) = {\left\{ \begin{array}{ll} \infty &{}\quad \xi < 0 \\ t(\frac{\xi ^2}{2} - \theta \xi ) &{}\quad \xi \ge 0 \end{array}\right. }. \end{aligned}$$

The next lemma implies existence of the rate functions which will appear when we use equation (12) to prove that \(J_{s,t}(x)\) satisfies a variational problem in the next subsection. Versions of this result appear in several other papers, so we elect not to re-prove it. The exact statement we need appears in [14].

Lemma 3.2

[14], Lemma 3.6] Suppose that for each \(n\), \(X_n\) and \(Y_n\) are independent, that the limits

$$\begin{aligned} \lambda (s)&= \lim _{n \rightarrow \infty } - \frac{1}{n} \log P \left( X_n \ge ns \right) , \qquad \phi (s) = \lim _{n \rightarrow \infty } - \frac{1}{n} \log P \left( Y_n \ge n s \right) \end{aligned}$$

exist, and that \(\lambda \) is continuous. If there exists \(a_\lambda \) and \(a_\phi \) so that \(\lambda (a_\lambda ) = \phi (a_\phi ) = 0\), then

$$\begin{aligned} \lim _{n \rightarrow \infty } - \frac{1}{n} \log P\left( X_n + Y_n \ge nr\right)&= {\left\{ \begin{array}{ll} \inf _{a_\lambda \le s \le r - a_\phi }\{\phi (r-s) + \lambda (s)\} &{}\quad r\ge a_\phi + a_\lambda \\ 0 &{}\quad r \le a_\phi + a_\lambda \end{array}\right. }\\&= \lambda \square \phi (r). \end{aligned}$$

We define rate functions corresponding to the two parts of the decomposition in (12) as follows: for \(a \in [0,t)\), \(u\in (0,s]\), \(v \in [0,s)\), and \(x \in \mathbb {R}\) set

$$\begin{aligned} G_{a,s,t}^\theta (x)&= - \lim \frac{1}{n} \log P \left( B(na,nt) - \theta n (t-a) + \log Z_{1,\lfloor n s \rfloor }(na,nt) \ge nx \right) , \nonumber \\ H_{u, v, s, t}^\theta (x)&= - \lim \frac{1}{n} \log P \left( - \log Z_0^\theta (nt) + \log Z_{\lfloor n u\rfloor }^\theta (0) + \log Z_{\lfloor nv \rfloor , \lfloor n s \rfloor }(0,nt) \ge nx \right) . \end{aligned}$$
(13)

Recall that \(\log Z_j^\theta (0) = \sum _{k=1}^j r_k^\theta (0)\) is measurable with respect to the sigma algebra \(\sigma (B(s), B_k(s) : 1 \le k \le j; s \le 0)\) and that for \(0 \le u < nt\), \(\log Z_{j,\lfloor ns \rfloor }(u,t)\) is measurable with respect to the sigma algebra \(\sigma (B_k(s_k) : j \le k \le \lfloor ns\rfloor , u \le s_j \le nt)\). Combining the independence of the environment with the computations above, Theorem 3.1 and Lemma 3.2 imply that \(G_{a,s,t}^\theta (x)\) and \(H_{u, v, s, t}^\theta (x)\) are well-defined. In particular, we immediately obtain

Corollary 3.3

For \(a\in [0,t)\) and \(u \in (0,s]\), and \(v \in [0,s)\)

$$\begin{aligned} G_{a,s,t}^\theta (x)&= R_{t - a}^\theta \square J_{s,t - a}(x) , \qquad H_{u,v,s,t}^\theta (x) = R_t^\theta \square U_u^\theta \square J_{s - v, t}(x). \end{aligned}$$

In order to show that (12) leads to a variational problem, we need some regularity on \(G_{a,s,t}^\theta (x)\) and \(H_{u,v,s,t}^\theta (x)\). The three results that follow are purely technical, so we defer their proofs to the ‘Regularity for the Stationary Right Tail Rate Functions’ section of Appendix 1. Lemma 3.4 gives a strong kind of local uniform continuity of \(H_{u,v,s,t}^\theta (x)\) and Lemma 3.5 gives the same for \(G_{a,s,t}^\theta (x)\). The difference between the two statements comes from Lemma 3.6, which shows that \(G_{a,s,t}^\theta (x)\) degenerates to infinity locally uniformly near \(a = t\).

Lemma 3.4

Fix \(\theta , s, t > 0\) and a compact set \(K \subseteq \mathbb {R}\). Then

$$\begin{aligned} \lim _{\delta , \gamma , \epsilon \downarrow 0} \sup _{\mathop {r_1, r_2 \in K : |r_1 - r_2|< \epsilon }\limits ^{a,b,b' \in [0,s] : |b-b'|< \delta }}\left\{ |H_{a,b,s,t + \gamma }^\theta (r_1) - H_{a,b',s,t}^\theta (r_2)| \right\} = 0. \end{aligned}$$

Lemma 3.5

Fix \(\theta , s, t > 0\) and \(0 < \delta \le t\) and a compact set \(K \subseteq \mathbb {R}\). Then

$$\begin{aligned} \lim _{\epsilon ,\gamma \downarrow 0} \sup _{\mathop { r_1, r_2 \in K : |r_1 - r_2|< \epsilon }\limits ^{a_1,a_2 \in [0,t-\delta ] : |a_1 - a_2| < \gamma }}\left\{ |G_{a_1,s,t}^\theta (r_1) - G_{a_2,s,t}^\theta (r_2)|\right\} = 0. \end{aligned}$$

Lemma 3.6

Fix \(\theta , s, t > 0\) and \(K \subset \mathbb {R}\) compact. Then

$$\begin{aligned} \liminf _{a \uparrow t} \inf _{x \in K} \left\{ G_{a,s,t}^\theta (x)\right\}&= \infty . \end{aligned}$$

3.2 Coarse Graining and the Variational Problem

Fix \(a \in [0,t)\) and \(0 < \delta \le t - a\). Then (12) implies the following lower bounds

$$\begin{aligned}&\log \left( n \int _{a}^{a + \delta } \frac{Z_0^\theta (nu)}{Z_0^\theta (nt)}Z_{1,\lfloor n s \rfloor }(nu,nt)du \right) \le \sum _{k=1}^{\lfloor n s \rfloor } r_k^\theta (nt), \end{aligned}$$
(14)
$$\begin{aligned}&-\log Z_0^\theta (nt) + \log Z_j^\theta (0) + \log Z_{j,\lfloor n s \rfloor }(0,nt) \le \sum _{k=1}^{\lfloor n s \rfloor } r_k^\theta (nt). \end{aligned}$$
(15)

For any partition \(\{a_i\}_{i=0}^{N}\) of \([0,t]\), we also have

$$\begin{aligned} \sum _{k=1}^{\lfloor ns \rfloor } r_k^\theta (nt)&\le \max _{0 \le i \le N-1}\left\{ \log \left( n \int _{a_i}^{a_{i+1}} \frac{Z_0^\theta (nu)}{Z_0^\theta (nt)}Z_{1,\lfloor n s \rfloor }(nu,nt)du \right) \right\} \nonumber \\&\vee \max _{1 \le j \le \lfloor ns \rfloor } \left\{ - \log Z_0^\theta (nt) \!+\! \log Z_j^\theta (0) \!+\! \log Z_{j,\lfloor n s \rfloor }(0,nt)\right\} + \log ( N + 1 + ns). \end{aligned}$$
(16)

Our goal is now to show that estimates (14), (15), and (16) above lead to a variational characterization of the right tail rate function \(J_{s,t}(x)\):

$$\begin{aligned} U_s^\theta (x)&= \min \left\{ \inf _{0 \le a < t} \left\{ G_{a,s,t}^\theta (x) \right\} , \inf _{0 \le a < s} \left\{ H_{a, a,s,t}^\theta (x)\right\} \right\} \nonumber \\&= \min \left\{ \inf _{0 \le a < t} \left\{ R_{t - a}^\theta \square J_{s,t - a}(x) \right\} , \inf _{0 \le a < s} \left\{ R_t^\theta \square U_a^\theta \square J_{s - a, t}(x)\right\} \right\} . \end{aligned}$$
(17)

To improve the presentation of the paper, we have moved some of the estimates in the proofs that follow to Appendix 2.

Lemma 3.7

Fix \(\theta > 0\), \((s,t) \in (0,\infty )^2\) and \(x \in \mathbb {R}\). Then

$$\begin{aligned} U_s^\theta (x)&\le \min \left\{ \inf _{0 \le a < t}\left\{ G_{a,s,t}^\theta (x)\right\} , \inf _{0 \le a < s} \left\{ H_{a, a,s,t}^\theta (x) \right\} \right\} . \end{aligned}$$

Proof

For \(a \in [0,s)\), taking \(j = \lfloor a n \rfloor \) in inequality (15) above immediately implies

$$\begin{aligned} U_s^\theta (x)&\le H_{a, a,s,t}^\theta (x). \end{aligned}$$
(18)

Fix \(\delta \in (0,t)\); then for all \(a \in [0, t - \delta )\) and all \(u \in [0, a+\delta ]\), we have

$$\begin{aligned} Z_{1,1}(nu, n(a+\delta )) Z_{1, \lfloor ns \rfloor }(n (a + \delta ), nt)\le Z_{1 ,\lfloor ns \rfloor }(nu, nt). \end{aligned}$$
(19)

It then follows that

$$\begin{aligned}&P\left( \log \left( n \int _{a}^{a + \delta } \frac{Z_0^\theta (nu)}{Z_0^\theta (nt)}Z_{1,\lfloor n s \rfloor }(nu,nt)du \right) \ge nx \right) \\&\qquad \ge P\left( \log Z_{1, \lfloor ns \rfloor }(n(a + \delta ), nt) + \log \frac{Z_0^\theta (n(a+\delta ))}{Z_0^\theta (nt)}\right. \\&\qquad \qquad \left. + \log \left( n \int _{a}^{a + \delta } \frac{Z_0^\theta (nu)}{Z_0^\theta (n(a+\delta ))}Z_{1,1}(nu,n(a+\delta ) du\right) \ge nx \right) . \end{aligned}$$

Fix \(\epsilon >0\). By independence of the Brownian environment, we find that

$$\begin{aligned} \frac{-1}{n} \log&P\left( \log \left( n \int _{a}^{a + \delta } \frac{Z_0^\theta (nu)}{Z_0^\theta (nt)}Z_{1,\lfloor n s \rfloor }(nu,nt)du \right) \ge nx \right) \nonumber \\&\le \frac{-1}{n}\log P\left( \log Z_{1, \lfloor ns \rfloor }(n(a + \delta ), nt) + \log \frac{Z_0^\theta (n(a+\delta ))}{Z_0^\theta (nt)} \ge n(x + \epsilon )\right) \end{aligned}$$
(20)
$$\begin{aligned}&\qquad + \frac{-1}{n} \log P\left( \log \left( n \int _{a}^{a + \delta } \frac{Z_0^\theta (nu)}{Z_0^\theta (n(a+\delta ))}Z_{1,1}(nu,n(a+\delta ) du\right) \ge -n\epsilon \right) . \end{aligned}$$
(21)

Applying the lower bound obtained by considering the minimum of the Brownian increments on the interval \([a, a+\delta ]\) allows us to show that as \(n \rightarrow \infty \) the probability in line (21) tends to one. Then taking \(\limsup \) and recalling inequality (14), we obtain

$$\begin{aligned} U_s^\theta (x)&\le G_{a + \delta , s,t}^\theta (x + \epsilon ). \end{aligned}$$
(22)

By Lemma 3.5, we may take \(\delta ,\epsilon \downarrow 0\) in (22). Optimizing over \(a\) in the resulting equation and in (18) gives the result. \(\square \)

Lemma 3.8

Fix \(\theta > 0\), \((s,t) \in (0,\infty )^2\) and \(x \in \mathbb {R}\). Then

$$\begin{aligned} U_s^\theta (x)&\ge \min \left\{ \inf _{0 \le a < t} \left\{ G_{a,s,t}^\theta (x) \right\} , \inf _{0 \le a < s} \left\{ H_{a,a,s,t}^\theta (x) \right\} \right\} . \end{aligned}$$

Proof

Fix a large \(p > 1\) and small \(\epsilon , \gamma > 0\). Consider uniform partitions \(\{a_i\}_{i=0}^{M}\) of \([0,t]\) and \(\{b_i\}_{i=0}^{N}\) of \([0,s]\) of mesh \(\nu = \frac{t}{M+1}\) and \(\delta = \frac{s}{N+1}\) respectively. We will add restrictions on these parameters later in the proof. Take \(n\) sufficiently large that \(\lfloor b_i n \rfloor < \lfloor b_{i+1} n\rfloor \) for all \(i\).

Fix \(j < \lfloor ns \rfloor \) not equal to any of the partition points \(\lfloor b_i n \rfloor \) and consider \(i\) so that \(\lfloor b_i n \rfloor < j < \lfloor b_{i+1} n \rfloor \). Notice that \(Z_0^\theta (nt)\) is \(\sigma (B(nt))\) measurable and \( Z_j^\theta (0)\) is measurable with respect to \(\sigma (B(s),B_1(s), \ldots , B_j(s) : s \le 0)\), so these random variables and \(Z_{j, \lfloor ns \rfloor }(u,v)\) are mutually independent if \(0\le u < v\). It follows from translation invariance and this independence that

$$\begin{aligned}&P\big (- \log Z_0^\theta (nt) + \log Z_j^\theta (0) + \log Z_{j, \lfloor ns \rfloor }(0,nt) \ge nx\big ) \\&\quad = P\left( - \log Z_0^\theta (nt) + \log Z_j^\theta (0) + \log Z_{j, \lfloor ns \rfloor }(n \gamma ,n(t + \gamma )) \ge nx \right) . \end{aligned}$$

We have

$$\begin{aligned} Z_{\lfloor b_i n \rfloor , \lfloor ns \rfloor }(0 , n(t + \gamma ))&\ge Z_{\lfloor b_i n \rfloor , j}(0 ,n \gamma ) Z_{j, \lfloor ns \rfloor }(n \gamma , n(t + \gamma )). \end{aligned}$$

It then follows that

$$\begin{aligned}&P\left( - \log Z_0^\theta (nt) + \log Z_j^\theta (0) + \log Z_{j, \lfloor ns \rfloor }(0,nt) \ge nx \right) \\&\qquad \le P\left( - \log Z_0^\theta (nt) + \log Z_{\lfloor b_{i+1} n \rfloor }^\theta (0) + \log Z_{\lfloor b_i n \rfloor , \lfloor ns \rfloor }(0 , n(t + \gamma )) \ge n (x - 2\epsilon ) \right) \\&\quad \qquad + P \left( \log Z_{\lfloor b_i n \rfloor , j}(0, n \gamma ) \le - n \epsilon \right) + P\left( \sum _{k= j+1}^{ \lfloor b_{i+1} n \rfloor } r_k^\theta (0) \le - n \epsilon \right) . \end{aligned}$$

Using the moment bound in Lemma 6.2 with \(\xi = -p\) for \(p > 1\) and the exponential Markov inequality gives the bound

$$\begin{aligned} P \left( \log Z_{\lfloor b_i n \rfloor , j}(0, n\gamma ) \le - n \epsilon \right)&\le e^{- n p\left( \epsilon - p \gamma - \delta \log \frac{\delta }{\gamma }\right) + o(n)} \le e^{- n\frac{\epsilon }{2} p + o(n)}. \end{aligned}$$

For the last inequality, we first require \(\gamma < \frac{\epsilon }{4p}\) and then take \(\delta \) small enough that \(\delta \log \frac{\delta }{\gamma } < \frac{\epsilon }{4}\). The exponential Markov inequality and the known moment generating function of the i.i.d. sum give the bound

$$\begin{aligned} P\left( \sum _{k= j+1}^{ \lfloor b_{i+1} n \rfloor } r_k^\theta (0) \le - n \epsilon \right)&\le e^{-n p\left( \epsilon - \delta p^{-1}\log \left( \Gamma (\theta + p) \Gamma (\theta )^{-1}\right) \right) } \le e^{- n p \frac{\epsilon }{2}} \end{aligned}$$

where in the last step we additionally require \(\delta < \frac{\epsilon p}{4} \log \left( \Gamma (\theta + p) \Gamma (\theta )^{-1}\right) ^{-1}\). For the case that \(j\) is a partition point, we have

$$\begin{aligned}&P\left( - \log Z_0^\theta (nt) + \log Z_{\lfloor b_i n \rfloor }^\theta (0) + \log Z_{ \lfloor b_i n \rfloor , \lfloor ns \rfloor }(0,nt) \ge nx \right) \\&\qquad \le P\left( - \log Z_0^\theta (nt) + \log Z_{\lfloor b_{i+1} n \rfloor }^\theta (0) +Z_{\lfloor b_i n \rfloor , \lfloor ns \rfloor }(0 , n(t + \gamma )) \ge n (x - 2\epsilon ) \right) \\&\qquad \qquad + P\left( \sum _{k= \lfloor b_i n \rfloor }^{ \lfloor b_{i+1} n \rfloor } r_k^\theta (0) \le - 2n \epsilon \right) . \end{aligned}$$

and the same error bound as above applies. We now turn to the problem of estimating the integral

$$\begin{aligned}&P \left( \log \left( n \int _{a_i}^{a_{i+1}} \frac{Z_0^\theta (nu)}{Z_0^\theta (nt)}Z_{1,\lfloor n s \rfloor }(nu,nt)du\right) \ge nx\right) \\&\qquad \le P \left( \log \left( \frac{Z_0^\theta (na_i)}{Z_0^\theta (nt)}Z_{1,\lfloor ns \rfloor }(na_i, nt)\right) \ge n (x - \epsilon ) \right) \\&\qquad \qquad + P\left( \log \left( n \int _{a_i}^{a_{i+1}} \frac{Z_0^\theta (nu)}{Z_0^\theta (n a_i)}\frac{Z_{1,\lfloor n s \rfloor }(nu,nt)}{Z_{1,\lfloor n s \rfloor }(n a_i,nt)}du \right) \ge n \epsilon \right) . \end{aligned}$$

By Lemma 6.5 we have

$$\begin{aligned} P\left( \log \left( n \int _{a_i}^{a_{i+1}} \frac{Z_0^\theta (nu)}{Z_0^\theta (n a_i)} \frac{Z_{1,\lfloor n s \rfloor }(nu,nt)}{Z_{1,\lfloor n s \rfloor }(n a_i,nt)}du \right) \ge n \epsilon \right)&\le \exp \left\{ -n \left( \frac{\epsilon - \theta \nu }{2\sqrt{\nu }}\right) ^2 +o(n)\right\} \end{aligned}$$

where we require \(\nu < \frac{\epsilon }{\theta }\).

Take \(n\) sufficiently large that \(\log (ns + N) \le n \epsilon \). It follows from (16) and union bounds that

$$\begin{aligned}&\frac{1}{n} \log P \left( \sum _{k=1}^{\lfloor ns \rfloor } r_k^\theta (nt) \ge nx \right) \le \frac{1}{n} \log (ns + N)\\&\qquad +\max _{0 \le i \le M-1}\left\{ \frac{1}{n} \log P \left( \log \left( n \int _{a_i}^{a_{i+1}} Z_0^\theta (nu) Z_0^\theta (nt)^{-1} Z_{1,\lfloor n s \rfloor }(nu,nt)du \right) \ge n(x- \epsilon ) \right) \right\} \\&\qquad \vee \max _{1 \le j \le \lfloor ns \rfloor }\left\{ \frac{1}{n} \log P\left( -\log Z_0^\theta (nt) + \log Z_j^\theta (0) + \log Z_{j,\lfloor n s \rfloor }(0,nt) \ge n(x - \epsilon ) \right) \right\} . \end{aligned}$$

Combining this with the previous estimates, multiplying by \(-1\) and sending \(n \rightarrow \infty \) gives

$$\begin{aligned} U_s^\theta (x)&\!\ge \! \min _{0 \le i \le M -1} \left\{ G_{a_i s, t}^\theta (x \!-\! 2 \epsilon )\right\} \wedge \left( \frac{\epsilon \!-\! \theta \nu }{2\sqrt{\nu }}\right) ^2 \!\wedge \! \frac{p \epsilon }{2} \!\wedge \! \min _{0 \le i \le N -1}\left\{ H_{b_{i+1}, b_i, s, t + \gamma }(x \!-\! 3 \epsilon )\right\} \\&\ge \inf _{a \in [0,t)}\left\{ G_{a,s,t}^\theta (x-2\epsilon )\right\} \wedge \left( \frac{\epsilon - \theta \nu }{2\sqrt{\nu }}\right) ^2 \wedge \frac{p \epsilon }{2} \\&\wedge \inf _{a \in [0,s)}\left\{ H_{a, a, s, t }(x) - \sup _{a,b,b' \in [0,s] : |b-b'|< \delta }\left\{ |H_{a,b,s,t + \gamma }^\theta (x -3 \epsilon ) - H_{a,b',s,t}^\theta (x)|\right\} \!\!\right\} . \end{aligned}$$

We first send \(\delta \downarrow 0\), then \(\gamma \downarrow 0\), then \(\nu \downarrow 0\), then \(p \uparrow \infty \). By Lemma 3.6, there is \(\eta > 0\) so that for all \(\epsilon \in [0,1]\), we have

$$\begin{aligned} \inf _{a \in [0,t )}\left\{ G_{a,s,t}^\theta (x-2\epsilon )\right\}&= \inf _{a \in [0,t - \eta ]}\left\{ G_{a,s,t}^\theta (x-2\epsilon )\right\} . \end{aligned}$$

Now, take \(\epsilon \downarrow 0\) and use Lemmas 3.4 and 3.5. This gives the desired bound

$$\begin{aligned} U_s^\theta (x)&\ge \min \left\{ \inf _{a \in [0,t)}\left\{ G_{a,s,t}^\theta (x)\right\} , \inf _{a \in [0,t)} \left\{ H_{a,a,s,t}^\theta (x)\right\} \right\} . \end{aligned}$$

\(\square \)

We now turn the variational problem for the right tail rate functions into a variational problem involving Legendre–Fenchel transforms.

Lemma 3.9

For any \(\theta > 0\) let \(\xi \in (0,\theta )\). Then \(J_{s,t}^*(\xi )\) satisfies the variational problem

$$\begin{aligned} 0 = \max \bigg \{&\sup _{0 \le a < t}\left\{ (t-a)\left( \frac{1}{2} \xi ^2 - \theta \xi \right) - s \log \frac{\Gamma (\theta - \xi )}{\Gamma (\theta )} + J_{s,t-a}^*(\xi )\right\} , \\&\sup _{0 \le a < s}\left\{ t\left( \frac{1}{2} \xi ^2 - \theta \xi \right) - (s-a) \log \frac{\Gamma (\theta - \xi )}{\Gamma (\theta )} + (J_{s - a, t})^*(\xi )\right\} \!\!\bigg \}. \end{aligned}$$

Proof

Lemmas 3.7 and 3.8 imply (17). Infimal convolution is Legendre–Fenchel dual to addition for proper convex functions [26], Theorem 16.4] so we find

$$\begin{aligned} (U_s^\theta )^*(\xi )&= \sup _{x \in \mathbb {R}} \left\{ \xi x - \min \left\{ \inf _{0 \le a < t}\left\{ R_{t - a}^\theta \square J_{s,t - a}(x)\right\} , \inf _{0 \le a < s}\left\{ R_t^\theta \square U_a^\theta \square J_{s - a, t}(x)\right\} \right\} \!\right\} \\&= \sup _{x \in \mathbb {R}}\left\{ \max \left\{ \sup _{0 \le a < t}\left\{ \xi x -R_{t - a}^\theta \square J_{s,t - a}(x)\right\} , \sup _{0 \le a < s}\left\{ \xi x - R_t^\theta \square U_a^\theta \square J_{s - a, t}(x)\right\} \right\} \!\right\} \\&= \max \left\{ \sup _{0 \le a < t} \left\{ (R_{t-a}^\theta )^*(\xi ) + J_{s,t-a}^*(\xi )\right\} ,\right. \nonumber \\&\quad \left. \sup _{0 \le a < s}\left\{ (R_t^\theta )^*(\xi ) + (U_a^\theta )^*(\xi ) + (J_{s - a, t})^*(\xi ) \right\} \right\} . \end{aligned}$$

If \(\xi \in (0, \theta )\), then \((U_s^\theta )^*(\xi ) < \infty \), so we may subtract \((U_s^\theta )^*(\xi )\) from both sides. Substituting in the known Legendre–Fenchel transforms gives the result. \(\square \)

3.3 Solving the Variational Problem

Next, we show that the variational problem in Lemma 3.9 identifies \(J_{s,t}^*(\xi )\) for \(\xi > 0\). To show the analogous result in [14], the authors followed the approach of rephrasing the variational problem as a Legendre–Fenchel transform in the space-time variables and appealing to convex analysis. We present an alternate method for computing \(J_{s,t}^*(\xi )\) in the next proposition, which has the advantage of allowing us to avoid some of the technicalities in that argument. This direct approach is the main reason we are able to appeal to the Gärtner-Ellis theorem to prove the large deviation principle.

Proposition 3.10

Let \(I \subseteq \mathbb {R}\) be open and connected and let \(h,g:I \rightarrow \mathbb {R}\) be twice continuously differentiable functions with \(h'(\theta ) > 0\) and \(g'(\theta ) < 0\) for all \(\theta \in I\). For \((x,y) \in (0,\infty )^2\), define

$$\begin{aligned} f_{x,y}(\theta ) = x h(\theta ) + y g(\theta ) \end{aligned}$$

and suppose that \(\frac{d^2}{d\theta ^2}f_{x,y}(\theta ) > 0\) for all \((x,y) \in (0,\infty )^2\) and that \(f_{x,y}(\theta ) \rightarrow \infty \) as \(\theta \rightarrow \partial I\) (which may be a limit as \(\theta \rightarrow \pm \infty \)). If \(\Lambda (x,y)\) is a continuous function on \((0,\infty )^2\) with the property that for all \((x,y) \in (0,\infty )^2\) and \(\theta \in I\) the identity

$$\begin{aligned} 0&= \sup _{0 \le a < x}\left\{ \Lambda (x-a, y) - f_{x-a,y}(\theta ) \right\} \vee \sup _{0 \le b < y}\left\{ \Lambda (x, y - b) - f_{x,y-b}(\theta ) \right\} \end{aligned}$$
(23)

holds, then

$$\begin{aligned} \Lambda (x,y)&= \min _{\theta \in I}\left\{ f_{x,y}(\theta )\right\} . \end{aligned}$$

Proof

Fix \((x,y) \in (0,\infty )^2\) and call \(\nu = \frac{y}{x}\). Under these hypotheses, there exists a unique \(\theta _{x,y}^* = \arg \min _{\theta \in I} f_{x,y}(\theta ) = \theta _{1,\nu }^ *\). Identity (23) implies that for all \(a \in [0,x)\) and \(b \in [0,y)\) we have

$$\begin{aligned} \Lambda (x-a,y) \le f_{x-a,y}(\theta _{x-a,y}^*), \qquad \Lambda (x,y-b) \le f_{x,y-b}(\theta _{x,y-b}^*), \end{aligned}$$

and therefore for any \(\theta \in I\), \(a \in [0,x)\) and \(b \in [0,y)\),

$$\begin{aligned} \Lambda (x-a,y) - f_{x-a,y}(\theta )&\le f_{x-a,y}(\theta _{x-a,y}^*) - f_{x-a,y}(\theta ), \end{aligned}$$
(24)
$$\begin{aligned} \Lambda (x,y-b) - f_{x,y-b}(\theta )&\le f_{x,y-b}(\theta _{x,y-b}^*) - f_{x, y-b}(\theta ). \end{aligned}$$
(25)

Uniqueness of minimizers implies that \(f_{x-a,y}(\theta _{x-a,y}^*) - f_{x-a,y}(\theta ) < 0\) unless \(\theta = \theta _{x-a,y}^*\) and similarly \(f_{x,y-b}(\theta _{x,y-b}^*) - f_{x, y-b}(\theta ) < 0\) unless \(\theta = \theta _{x,y-b}^*\). Notice that \(\theta _{1,\nu }^*\) solves

$$\begin{aligned} 0&= h'(\theta _{1,\nu }^*) + \nu g'(\theta _{1,\nu }^*). \end{aligned}$$
(26)

By the implicit function theorem, we may differentiate the previous expression with respect to \(\nu \) to obtain

$$\begin{aligned} \frac{d \theta _{1,\nu }^*}{d\nu }&= - \frac{g'(\theta _{1,\nu }^*)}{h''(\theta _{1,\nu }^*) + \nu g''(\theta _{1,\nu }^*)} > 0. \end{aligned}$$
(27)

Now, set \(\theta = \theta _{x,y}^*\) in (23). Equality (27) implies that for \(a \in (0,x)\) and \(b \in (0,y)\), \(\theta _{(x,y-b)}^* < \theta _{(x,y)}^* < \theta _{(x-a,y)}^*\). Then (24) and (25) give us the inequalities

$$\begin{aligned} \Lambda (x-a,y) - f_{x-a,y}(\theta _{x,y}^*)&\le f_{x-a,y}(\theta _{x-a,y}^*) - f_{x-a,y}(\theta _{x,y}^*) < 0, \end{aligned}$$
(28)
$$\begin{aligned} \Lambda (x,y-b) - f_{x,y-b}(\theta _{x,y}^*)&\le f_{x,y-b}(\theta _{x,y-b}^*) - f_{x, y-b}(\theta _{x,y}^*) < 0. \end{aligned}$$
(29)

Notice that (23) implies either there exists \(a_n \rightarrow a \in [0,x]\) or \(b_n \rightarrow b \in [0,y]\) so that one of the following hold:

$$\begin{aligned} \Lambda (x-a_n,y) - f_{x-a_n,y}(\theta _{x,y}^*) \rightarrow 0, \qquad \Lambda (x,y-b_n) - f_{x,y-b_n}(\theta _{x,y}^*) \rightarrow 0. \end{aligned}$$

Our goal is to show that the only possible limits are \(a_n \rightarrow 0\) or \(b_n \rightarrow 0\), from which the result follows from continuity. Continuity and inequalities (28) and (29) rule out the possibilities \(a \in (0,x)\) and \(b \in (0,y)\) respectively. It therefore suffices to show that

$$\begin{aligned}&\limsup _{a \rightarrow x^-} f_{x-a,y}(\theta _{x-a,y}^*) - f_{x-a,y}(\theta _{x,y}^*) <0, \end{aligned}$$
(30)
$$\begin{aligned}&\limsup _{b \rightarrow y^-} f_{x,y-b}(\theta _{x,y-b}^*) - f_{x, y-b}(\theta _{x,y}^*) < 0. \end{aligned}$$
(31)

We will only write out the proof of (30), since the proof of (31) is similar. For any fixed \(a \in (0,x)\), we have

$$\begin{aligned} f_{x-a,y}(\theta _{x-a,y}^*) - f_{x-a,y}(\theta _{x,y}^*) < 0. \end{aligned}$$

It suffices to show that the previous expression is decreasing in \(a\). Differentiating the previous expression and using (26) and the fact that \(\theta _{(x,y)}^* < \theta _{(x-a,y)}^*\), we find

$$\begin{aligned}&\frac{d}{da}\Bigg ( (x-a) h(\theta _{(x-a,y)}^*) + y g(\theta _{(x-a,y)}^*) - \Bigg [(x-a) h(\theta _{(x,y)}^*) + y g(\theta _{(x,y)}^*) \Bigg ]\Bigg ) \\&\qquad = h(\theta _{(x,y)}^*) - h(\theta _{(x-a,y)}^*) < 0. \end{aligned}$$

\(\square \)

Corollary 3.11

For all \(\xi > 0\),

$$\begin{aligned} J_{s,t}^*(\xi )&= \min _{\theta > \xi }\left\{ t\left( - \frac{\xi ^2}{2} + \theta \xi \right) + s \log \frac{\Gamma (\theta - \xi )}{\Gamma (\theta )}\right\} \\&= \min _{\mu > 0}\left\{ t\left( \frac{\xi ^2}{2} + \xi \mu \right) - s \log \frac{\Gamma (\mu + \xi )}{\Gamma (\mu )} \right\} . \end{aligned}$$

Proof

It follows from the variational representation in Lemma 3.9 that \(J_{s,t}^*(\xi )\) is not infinite for any choice of the parameters \(\xi ,s,t>0\). It then follows from Lemma 5.4 and [26], Theorem 10.1] that \(J_{s,t}^*(\xi )\) is continuous in \((s,t) \in (0,\infty )^2\).

Fix \(\xi \) and set \(I = \{ \theta : \theta > \xi \}\). For \(\theta \in I\) and \(s,t \in (0,\infty )\), define

$$\begin{aligned}&h(\theta ) = - \frac{\xi ^2}{2} + \theta \xi ,&g(\theta ) = \log \frac{\Gamma (\theta - \xi )}{\Gamma (\theta )}, \\&f_{s,t}(\theta ) = sh(\theta ) + tg(\theta ),&\Lambda (s,t) = J_{s,t}^*(\xi ). \end{aligned}$$

Lemma 3.9 shows that with these definitions \(J_{s,t}^*(\xi )\) solves the variational problem in Proposition 3.10. Because \(\Psi _1(x) > 0\) and \(\Psi _2(x) < 0\), we see that for \(\theta \in I\)

$$\begin{aligned}&g'(\theta ) = \Psi _0(\theta - \xi ) - \Psi _0(\theta ) < 0,&g''(\theta ) = \Psi _1(\theta - \xi ) - \Psi _1(\theta ) > 0. \end{aligned}$$

It then follows that \(\frac{d^2}{d\theta ^2}f_{s,t}(\theta ) > 0\). Moreover, since \(\log \frac{\Gamma (\theta - \xi )}{\Gamma (\theta )}\) grows like \(-\xi \log (\theta )\) at infinity and \(-\log (\theta - \xi )\) at \(\xi \), \(f_{s,t}(\theta )\) also tends to infinity at the boundary of \(I\) and the result follows.

The second equality is the substitution \(\mu = \theta - \xi \). \(\square \)

4 Moment Lyapunov Exponents and the LDP

The next result would be Varadhan’s theorem if \(J_{s,t}(x)\) were a full rate function, rather than a right tail rate function. The proof is somewhat long and essentially the same as the proof of Varadhan’s theorem, so we omit it. Details of a similar argument for the stationary log-gamma polymer can be found in [14], Lemma 5.1]. The exponential moment bound needed for the proof follows from Lemma 6.2.

Lemma 4.1

For \(\xi > 0\),

$$\begin{aligned} J_{s,t}^*(\xi )&= \lim _{n \rightarrow \infty } \frac{1}{n} \log E \left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)} \right] \end{aligned}$$

and in particular the limit exists.

Remark 4.2

Lemma 4.1 shows that \(J_{s,t}^*(\xi )\) is the \(\xi \) moment Lyapunov exponent for the parabolic Anderson model associated to this polymer. With this in mind, the second formula in the statement of Corollary 3.11 above agrees with the conjecture in [7], Appendix A.1].

To see this, we first observe that the partition function we study differs slightly from the partition function \(Z_\beta (t,n)\) studied in [7] (defined in equation (3) of that paper). As we saw was the case for \(Z_n(\beta )\) in equation (3), up to normalization constants both \(Z_{0,\lfloor ns \rfloor }(0,nt)\) and \(Z_\beta (t,n)\) are conditional expectations of functionals of a Poisson path. The normalization constant for \(Z_{0, \lfloor ns \rfloor }(0,nt)\) is given by the Lebesgue measure of the Weyl chamber \(A_{\lfloor ns \rfloor +1, nt}\), while the normalization constant for \(Z_\beta (t,n)\) is \(P_{\pi (0)=0}\left( \pi (t) = n\right) \) where \(\pi (\cdot )\) is again a rate one Poisson process. There is a further difference in that [7] adds a pinning potential of strength \(\frac{\beta }{2}\) at the origin to the definition of \(Z_\beta (t,n)\), which introduces a multiplicative factor of \(e^{-\frac{\beta }{2}t}\). Combining these changes and restricting to the parameters studied in [7], Appendix A.1], we have the relation

$$\begin{aligned} e^{- \frac{n}{2}}\frac{P_{\pi (0) = 0}\left( \pi (n) = \lfloor n \nu \rfloor \right) }{|A_{\lfloor n \nu \rfloor + 1,n}|}Z_{0, \lfloor n \nu \rfloor }(0,n)&= Z_1(n, \lfloor n \nu \rfloor ). \end{aligned}$$

Since \(P_{\pi (0) = 0}\left( \pi (n) = \lfloor n \nu \rfloor \right) |A_{\lfloor n \nu \rfloor +1,n}|^{-1} = e^{-n}\), Corollary 3.11 and Lemma 4.1 then imply that for any \(k > 0\),

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n} \log E\left[ Z_1(n, \lfloor n \nu \rfloor )^k \right]&= - \frac{3}{2}k + \min _{z > 0}\left\{ \frac{k^2}{2} + k z - \nu \log \frac{\Gamma (z + k)}{\Gamma (z)} \right\} \\&= \min _{z > 0}\left\{ \frac{k(k-3)}{2} + k z- \nu \log \frac{\Gamma (z + k)}{\Gamma (z)} \right\} , \end{aligned}$$

which is the extension of the moment Lyapunov exponent \(H_k(z_k^0)\) conjectured in the middle of page 24 of [7].

Our next goal is to show that the left tail large deviations are not relevant at the scale we consider. This proof is based on the proof of [14], Lemma 4.2] which contains a small mistake; as currently phrased, the argument in that paper only works for \(s,t \in \mathbb {Q}\). This problem can be fixed by altering the geometry of the proof, but doing this adds some technicalities which can be avoided in the model we study. We will follow an argument similar to the original proof for \(s \in \mathbb {Q}\), then show that this implies what we need for all \(s\).

Proposition 4.3

Fix \(s,t >0\). For all \(\epsilon > 0\)

$$\begin{aligned} \liminf _{n \rightarrow \infty } - \frac{1}{n} \log P\left( \log Z_{1, \lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon ) \right)&= \infty . \end{aligned}$$

Proof

First we consider the case \(s \in \mathbb {Q}\). There exists \(M \in \mathbb {N}\) large enough that \(M(s \wedge t) \ge 1\) and for all \(m \ge M\) we have

$$\begin{aligned} \frac{1}{m} E \log Z_{1, \lfloor ms \rfloor }(0,mt) \ge \rho (s,t) - \epsilon . \end{aligned}$$

Fix \(m \ge M\) so that \(ms \in \mathbb {N}\). We will denote coordinates in \(\mathbb {R}^{\lfloor ns \rfloor - 1}\) by \((u_1, \ldots , u_{\lfloor ns \rfloor -1})\). For \(a,b,s,t \in (0,\infty )\) and \(n,k,l \in \mathbb {N}\), define a family of sets \(A_{k,a}^{l,b} \subset \mathbb {R}^{\lfloor ns \rfloor - 1}\) by

$$\begin{aligned} A_{k,a}^{l,b}&= \{0 < u_1 < \cdots < u_{k-1} < a < u_k < \cdots < u_{k + l -1} \nonumber \\&\qquad < a + b < u_{k+l} < \cdots < u_{\lfloor ns \rfloor - 1} < nt\}. \end{aligned}$$

For \(j,k, \in \mathbb {Z}^+\), set

$$\begin{aligned} A_j^k&\equiv A_{j \lfloor ms \rfloor + 1, (j + k) mt}^{\lfloor ms \rfloor , mt}. \end{aligned}$$

For each \(n\) sufficiently large that the expression below is greater than one, define

$$\begin{aligned} N&= \left\lfloor \frac{n}{m} - \lfloor \sqrt{n} \rfloor - 2 \right\rfloor , \end{aligned}$$

so that we have

$$\begin{aligned} (n - 2m)t&\le (N + \lfloor \sqrt{n}\rfloor + 1)mt \le (n-m)t, \end{aligned}$$
(32)
$$\begin{aligned} \left( \lfloor \sqrt{n} \rfloor + 1\right) ms - 1&\le \lfloor ns \rfloor - N \lfloor ms \rfloor \le \left( \lfloor \sqrt{n}\rfloor + 2\right) ms. \end{aligned}$$
(33)

With this choice of \(N\), for \(0 \le k \le \lfloor \sqrt{n} \rfloor \) and \(0 \le j \le N-1\), \(A_j^k\) is nonempty. Then for \(0 \le k \le \lfloor \sqrt{n} \rfloor \), define

$$\begin{aligned} D_k&= \cap _{j=0}^{N-1} A_j^k \\&\quad \cap \left\{ u : 0 < u_1 \!<\! \cdots \!<\! u_{(N+1)ms - 1} \!<\! t\left( n - \frac{m}{2}\right) \!<\! u_{(N+1)ms} < \cdots < u_{\lfloor ns \rfloor - 1} < nt \right\} . \end{aligned}$$

To simplify the formulas that follow, we introduce the notation \(s_j = j ms\) and \(t_j^k = (j+k)mt\). In words, we can think of \(D_k\) as the collection of paths from \(0\) to \(nt\) which traverse the bottom line until \(t_0^k\), then for \(0 \le j \le N-1\) move from \(t_j^k\) to \(t_{j+1}^k\) along the next \(ms\) lines. The path then moves from \(t_N^k\) to \(t\left( n - \frac{m}{2}\right) \) along the next \(ms\) lines and finally proceeds to \(nt\) along the remaining lines. Observe that \(\{D_k\}_{k=0}^{\lfloor \sqrt{n} \rfloor }\) is a pairwise disjoint, non-empty family of sets. With the convention \(u_0 = 0\) and \(u_{\lfloor ns \rfloor } = nt\), we have the bound

$$\begin{aligned} Z_{1, \lfloor ns \rfloor }(0,nt)&\ge \sum _{k=0}^{\lfloor \sqrt{n}\rfloor } \int _{D_k} e^{\sum _{i=1}^{\lfloor ns \rfloor } B_i(u_{i-1}, u_{i})} du_1 \ldots u_{\lfloor ns \rfloor -1}. \end{aligned}$$

In the integral over \(D_k\), for each \(0 \le j \le N\) we add and subtract \(B_{s_j}(t_j^k)\) in the exponent. Similarly, add and subtract \(B_{s_{N+1}}\left( t\left( n - \frac{m}{2}\right) \right) \). The reason for this step is that this will make the product of integrals coming from the definition of \(D_k\) into a product of partition functions, as when we showed supermultiplicativity of the partition function in (7). Introduce

$$\begin{aligned} H_k^n&= \inf _{t_N^k = u_0 < u_1 < \cdots < u_{ms - 1} < u_{ms} = n\left( t - \frac{m}{2}\right) }\left\{ \sum _{i=0}^{ms-1}B_{s_N + i}(u_{i-1}, u_i)\right\} \end{aligned}$$

and observe that \(H_0^n \le B_{s_N}(t_N^0, t_N^k) + H_k^n\). Let \(C> 0\) be a uniform lower bound in \(n\) (recall that \(m\) is fixed) on the Lebesgue measure of the Weyl chamber in the definition of \(H_{\lfloor \sqrt{n} \rfloor }^n\). Such a bound exists by (32). Set \(I_n = \max _{t_{N-1}^0 \le u \le t_{N-1}^{\lfloor \sqrt{n}\rfloor }}\{B_{s_N}(t_{N-1}^0, u)\}\). We have the lower bound

$$\begin{aligned}&Z_{1, \lfloor ns \rfloor }(0,nt) \ge C Z_{s_{N+1}, \lfloor ns \rfloor }\left( t\left( n - \frac{m}{2}\right) , nt\right) \\&\quad e^{H_0^n - I_n}\left( \sum _{k=0}^{\lfloor \sqrt{n} \rfloor } e^{B_0(0, t_0^k)} \prod _{j=0}^{N-1}Z_{s_j,s_{j+1}}\left( t_j^k, t_{j+1}^k\right) \right) . \end{aligned}$$

We therefore have the upper bound

$$\begin{aligned}&P\left( \log Z_{1, \lfloor ns \rfloor }(0,nt) \le -n(\rho (s,t) - 6\epsilon ) \right) \\&\qquad \le P\left( \log Z_{(N+1)ms, \lfloor ns \rfloor }\left( t\left( n - \frac{m}{2}\right) , nt\right) \le -n \epsilon - \log C \right) \\&\qquad \qquad +P\left( \max _{0 \le k \le \lfloor \sqrt{n} \rfloor } \sum _{j=0}^{N-1}\log Z_{s_j + 1,s_{j+1}}\left( t_j^k, t_{j+1}^k\right) \le -n(\rho (s,t) - 2\epsilon ) \right) \\&\qquad \qquad + P\left( H_0 \le -n \epsilon \right) + P\left( \min _k B_0(t_0^k) \le -n\epsilon \right) + P\left( I_n \ge n \epsilon \right) . \end{aligned}$$

It follows from translation invariance, Lemma 6.6, and (33) that

$$\begin{aligned} P\left( \log Z_{(N+1)ms, \lfloor ns \rfloor }\left( t\left( n - \frac{m}{2}\right) , nt\right) \le -n \epsilon - \log \frac{mt}{12} \right) = O\left( e^{-n^\frac{3}{2}}\right) . \end{aligned}$$

We have

$$\begin{aligned}&P\left( \max _{0 \le k \le \lfloor \sqrt{n} \rfloor } \left\{ \sum _{j=0}^{N-1}\log Z_{s_j + 1,s_{j+1}}\left( t_j^k, t_{j+1}^k\right) \right\} \le -n(\rho (s,t) - 2\epsilon ) \right) \\&\qquad \qquad = P\left( \sum _{j=0}^{N-1}\log Z_{s_j + 1,s_{j+1}}\left( t_j^1, t_{j+1}^1\right) \le -n(\rho (s,t) - 2\epsilon ) \right) ^{\lfloor \sqrt{n}\rfloor } = O\left( e^{- c_1 n^{\frac{3}{2}}}\right) \end{aligned}$$

for some \(c_1 > 0\). The first equality comes from the fact that the terms in the maximum are i.i.d. and the second comes from large deviation estimates for an i.i.d. sum once we recall that \(N = \frac{n}{m} + o(n)\).

Recall that by (32), \(n\left( t - \frac{m}{2}\right) - t_N^0 = O(\sqrt{n})\). It follows from Lemma 6.4 that there exist \(c_2, C_2 > 0\) so that

$$\begin{aligned} P\left( H_0 \le -n \epsilon \right)&\le C_2 e^{- c_2 n^{\frac{3}{2}}}. \end{aligned}$$

The remaining two terms can be controlled with the reflection principle and are \(O\left( e^{-\frac{1}{2}n^{\frac{3}{2}}}\right) \).

Now let \(s\) be irrational. For each \(k\), fix \(\tilde{s}_k < s\) rational with \(e^{-k} < |\tilde{s}_k - s| < 2 e^{-k}\) and set \(\tilde{t}_k = t -\frac{1}{k} \). Call \(\alpha _k = s - \tilde{s}_k\) and \(\beta _k = t - \tilde{t}_k = \frac{1}{k}\). Subadditivity gives

$$\begin{aligned}&P\left( \log Z_{1, \lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon ) \right) \\&\qquad \le P\left( \log Z_{1, \lfloor n \tilde{s}_k \rfloor }(0,n\tilde{t}_k) \le n\left( \rho (\tilde{s}_k,\tilde{t}_k) - \frac{\epsilon }{2}\right) \right) \\&\qquad \qquad + P\left( \log Z_{\lfloor n \tilde{s}_k\rfloor , \lfloor ns \rfloor }(n \tilde{t}_k, nt) \le n\left( \rho (s,t) - \rho (\tilde{s}_k, \tilde{t}_k) - \frac{\epsilon }{2}\right) \right) . \end{aligned}$$

Since \(\tilde{s}_k\) is rational, we have already shown that the first term is negligible. Take \(k\) sufficiently large that \(\rho (s,t) - \rho (\tilde{s}_k, \tilde{t}_k) - \frac{\epsilon }{2} < -\frac{\epsilon }{4}\). By Lemma 6.3, we find

$$\begin{aligned}&\liminf _{n \rightarrow \infty } - \frac{1}{n} P\left( \log Z_{1, \lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon ) \right) \nonumber \\&\quad \ge \alpha _k J_{GUE}\left( \frac{\frac{\epsilon }{4} - \alpha _k \log \beta _k - \alpha _k + \alpha _k \log \alpha _k}{2\sqrt{\alpha _k \beta _k}} \right) . \end{aligned}$$

Using formula (38), \(J_{GUE}(r) = 4 \int _0^r \sqrt{x(x+2)}dx\), it is not hard to see that as \(k \rightarrow \infty \), this lower bound tends to infinity. \(\square \)

Lemma 4.4

Fix \(s,t >0\) and \(\xi < 0\). Then

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right]&= \xi \rho (s,t). \end{aligned}$$

Proof

Fix \(\epsilon > 0\) and small and recall that Lemma 6.2 and Jensen’s inequality imply that for any \(\xi < 0\), \(\sup _n\left\{ \frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right] \right\} < \infty \). The lower bound is immediate from

$$\begin{aligned} \frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right]&\ge \frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}1_{\{\log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) + \epsilon )\}}\right] \\&\ge \xi (\rho (s,t) + \epsilon ) + \frac{1}{n} \log P(\log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) + \epsilon )) \end{aligned}$$

once we recall that \(P(\log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) + \epsilon )) \rightarrow 1\).

For the upper bound, we decompose the expectation as follows

$$\begin{aligned} E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right]&=E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}1_{\{\log Z_{1,\lfloor ns \rfloor }(0,nt) > n(\rho (s,t) - \epsilon )\}}\right] \\&\quad \,+ E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}1_{\{\log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon )\}}\right] . \end{aligned}$$

Recalling that \(P\left( \log Z_{1,\lfloor ns \rfloor }(0,nt) > n(\rho (s,t) - \epsilon )\right) \rightarrow 1\), this leads to

$$\begin{aligned} \limsup _{n \rightarrow \infty }&\frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right] \\&\le \max \Big \{ \xi (\rho (s,t) - \epsilon ), \limsup _{n \rightarrow \infty } \frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right. \nonumber \\&\quad \times \left. 1_{\{\log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon )\}}\right] \Big \}. \end{aligned}$$

By Cauchy–Schwarz and Proposition 4.3

$$\begin{aligned} \limsup _{n \rightarrow \infty }&\frac{1}{n} \log E\left[ e^{\xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}1_{\{\log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon )\}}\right] \\&\le \frac{1}{2} \sup _n \left\{ \frac{1}{n} \log E\left[ e^{2 \xi \log Z_{1, \lfloor ns \rfloor }(0,nt)}\right] \right\} \\&\quad + \limsup _{n \rightarrow \infty }\frac{1}{2n} \log P\left( \log Z_{1,\lfloor ns \rfloor }(0,nt) \le n(\rho (s,t) - \epsilon )\right) = -\infty . \end{aligned}$$

\(\square \)

Combining the previous results, we are led to the proof of Theorem 2.2, from which we immediately deduce Theorem 2.3.

Proof of Theorem 2.2

Lemmas 4.1 and 4.4 give the limit for \(\xi \ne 0\) and the limit for \(\xi = 0\) is zero.

Note that \(\Lambda _{s,t}(\xi )\) is differentiable for \(\xi < 0\) the left derivative at zero is \(\rho (s,t)\). For \(\xi > 0\), there is a unique \(\mu (\xi )\) solving

$$\begin{aligned} \Lambda _{s,t}(\xi )&= t\left( \frac{\xi ^2}{2} + \xi \mu (\xi )\right) - s \log \frac{\Gamma (\mu (\xi ) + \xi )}{\Gamma (\mu (\xi ))}. \end{aligned}$$
(34)

This \(\mu (\xi )\) is given by the unique solution to

$$\begin{aligned} 0&= t \xi + s\left( \Psi _0(\mu (\xi )) - \Psi _0(\mu (\xi ) + \xi )\right) , \end{aligned}$$
(35)

which can be rewritten as

$$\begin{aligned} \frac{1}{\xi }\left( \Psi _0(\mu (\xi ) + \xi ) - \Psi _0(\mu (\xi )) \right)&= \frac{t}{s}. \end{aligned}$$

By the mean value theorem, there exists \(x \in [0,\xi )\) so that

$$\begin{aligned} \Psi _1^{-1}\left( \frac{t}{s}\right) - x&= \mu (\xi ). \end{aligned}$$

Using this, we see that \(\Lambda _{s,t}(\xi )\) is continuous at \(\xi = 0\). The implicit function theorem implies that \(\mu (\xi )\) is smooth for \(\xi > 0\). Differentiating (34) with respect to \(\xi \) and applying (35), we have

$$\begin{aligned} \frac{d}{d\xi } \Lambda _{s,t}(\xi )&= t\left( \xi + \mu (\xi ) \right) - s \Psi _0(\mu (\xi ) + \xi ). \end{aligned}$$

Substituting in for \(\mu (\xi )\), appealing to continuity, and taking \(\xi \downarrow 0\) gives

$$\begin{aligned} \lim _{\xi \downarrow 0} \frac{d}{d\xi } \Lambda _{s,t}(\xi )&= t \Psi _1^{-1}\left( \frac{t}{s}\right) - s \Psi _0\left( \Psi _1^{-1} \left( \frac{t}{s}\right) \right) \\&= \rho (s,t) \end{aligned}$$

which implies differentiability at zero and hence at all \(\xi \). \(\square \)

Proof of Theorem 2.3

The large deviation principle holds by Theorem 2.2 and the Gärtner-Ellis theorem [13], Theorem 2.3.6]. \(\square \)