44. chen.pu:26:two-time
^^^^^^^^^^^^^^^^^^^^^^^^^

.. centered:: Two-time spatial decorrelation for the flat KPZ fixed point
.. centered:: Le Chen and Fei Pu

**Abstract:** We establish quantitative two-time spatial decorrelation for the
Kardar--Parisi--Zhang fixed point with flat initial data. For every
:math:`s,t>0`, there exist constants :math:`C,c>0` such that

.. math::

   \left|\operatorname{Cov}(h(t,x),h(s,0))\right|
   \leq C\exp\{-c|x|^3\}, \qquad |x|\geq 1.

Unlike the fixed-time covariance, which is governed directly by the
Airy\ :sub:`1` process, the two-time covariance involves the nonlinear
variational evolution of the entire earlier height profile. Our proof combines
cubic-exponential mixing of the Airy\ :sub:`1` process with a uniform
localization estimate for intermediate optimizers in the directed landscape.
As a consequence, the centered spatial averages, normalized by
:math:`N^{1/2}`, converge in finite-dimensional distributions to a centered
Gaussian process whose covariance is the space-integrated two-time correlation
of the flat KPZ fixed point.

`Preprint <https://arxiv.org/abs/2607.17113>`_

:cite:`chen.pu:26:two-time` Le Chen & Fei Pu (2026) 'Two-time spatial
decorrelation for the flat KPZ fixed point', *preprint arXiv:2607.17113, 24
pages*

.. code-block:: bibtex

   @article{chen.pu:26:two-time,
      title         = {Two-time spatial decorrelation for the flat {KPZ} fixed point},
      author        = {Le Chen and Fei Pu},
      year          = {2026},
      month         = {July},
      journal       = {Preprint arXiv:2607.17113},
      url           = {http://arXiv.org/abs/2607.17113}
   }


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