Abstract
AbstractWe study the decay of the covariance of the $$\hbox {Airy}_1$$
Airy
1
process, $${{\mathcal {A}}}_1$$
A
1
, a stationary stochastic process on $$\mathbb {R}$$
R
that arises as a universal scaling limit in the Kardar–Parisi–Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $${{\,\textrm{Cov}\,}}({{\mathcal {A}}}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$$
Cov
(
A
1
(
0
)
,
A
1
(
u
)
)
=
e
-
(
4
3
+
o
(
1
)
)
u
3
as $$u\rightarrow \infty $$
u
→
∞
. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $$\mathcal{A}_1$$
A
1
to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the $$\hbox {Airy}_1$$
Airy
1
process together with the FKG inequality.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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