Upper Tail Bounds for Stationary KPZ Models

Abstract

We present a proof of an upper tail bound of the correct order (up to a constant factor in the exponent) in two classes of stationary models in the KPZ universality class. The proof is based on an exponential identity due to Rains in the case of last passage percolation with exponential weights, and recently re-derived by Emrah–Janjigian–Seppäiläinen (EJS). Our proof follows very similar lines for the two classes of models we consider, using only general monotonocity and convexity properties, and can thus be expected to apply to many other stationary models.

A Tail Estimates from Moment Generating Functions

Proposition A.1

Let X be a random variable, and constants $c, C, \delta >0$ and $N \ge 1$ such that the estimates,

$$\begin{aligned} c \mathrm {e}^{ c N a^3} \le \mathbb {E}\left[ \mathrm {e}^{ a X} \right] \le C \mathrm {e}^{ C N a^3} \end{aligned}$$

(48)

hold for all $a \le \delta N$. Then there are constants $c', C'>0$ and $\delta ' >0$ depending only on $c, C, \delta$ so that

$$\begin{aligned} c' \mathrm {e}^{ - C' u^{3/2} N^{-1/2} } \le \mathbb {P}\left[ X > u \right] \le C' \mathrm {e}^{ - c' u^{3/2} N^{-1/2} } \end{aligned}$$

(49)

for $0< u < \delta ' N$. If only the upper bound holds in (48) then the upper bound still holds in (49).

Proof

The upper bounds follows from Markov’s inequality. For the lower bound, let $u_0 >0$. Then let $\delta> 2 a >0$ and $0< u_0 < u_1$. Then,

$$\begin{aligned} c \mathrm {e}^{ c N a^3} \le \mathbb {E}[ \mathrm {e}^{a X} ] \le \mathrm {e}^{ a u_1} \mathbb {P}[ X> u_0] + \mathrm {e}^{ a u_0} + \mathbb {E}[ \mathrm {e}^{2 a X} ]^{1/2} \mathbb {P}[ X > u_1 ]^{1/2} \end{aligned}$$

Choosing $a = C' (u_0 / N)^{1/2}$ for some large $C' >0$ (and assuming $u_0$ sufficiently small so that the requirement $2a < \delta$ is respected) we see that

$$\begin{aligned} c \mathrm {e}^{ c N a^3} \le \mathbb {E}[ \mathrm {e}^{a X} ] \le \mathrm {e}^{ a u_1} \mathbb {P}[ X> u_0]+ \mathbb {E}[ \mathrm {e}^{2 a X} ]^{1/2} \mathbb {P}[ X > u_1 ]^{1/2} \end{aligned}$$

for some new $c>0$. By our upper bounds and choice of a,

$$\begin{aligned} \mathbb {E}[ \mathrm {e}^{2 a X} ]^{1/2} \mathbb {P}[ X > u_1 ]^{1/2} \le C \mathrm {e}^{ C C'^3 u_0^{3/2} N^{-1/2}} \mathrm {e}^{ - c u_1^{3/2} N^{-1/2}}. \end{aligned}$$

Taking $u_1 = C'' u_0$ for some large $C'' >0$ we see that,

$$\begin{aligned} c \mathrm {e}^{ c N a^3} \le \mathrm {e}^{ a u_1} \mathbb {P}[ X > u_0]. \end{aligned}$$

This yields the claim. $\square$