Abstract:  In the study of (1+1)-dimensional random growth processes, a foundational model is the longest-increasing-subsequence problem: from n independent uniform points in the unit square, find the longest chain whose horizontal and vertical coordinates are both increasing. It is known that the maximal length is asymptotically 2n^(1/2), and the longest chain concentrates near the diagonal line y=x. In this talk, we add a geometric constraint: the chain must be convex. I will discuss the history of this problem, as well as recent joint work with Arnab Sen.

Abstract: The four-corner Cantor set is a planar analogue of the classical Cantor set and arises in several areas of analysis, including the study of Kakeya sets and removable singularities for analytic functions. A central problem is to understand how this set behaves when projected onto lines. This turns out to be a very difficult question, so we study a random variant of the Cantor set, where we are able to obtain sharp estimates. This is joint work with Pablo Shmerkin and Ville Suomala. 

Abstract:  In directed LPP on the square lattice with iid exponential weights, the coalescence time of two semi-infinite geodesics with a fixed asymptotic direction has a heavy tail with KPZ exponent 2/3, and hence infinite expectation. We show that three geodesics behave differently. In some models of directed first- and last-passage percolation with general iid weights, suppose we are given a stationary, non-crossing family of semi-infinite geodesics (such as those arising from Busemann functions). Consider three geodesics started from colinear points spaced distance k apart. We show that for every k, the first time at which some pair among the three coalesces has finite expectation. This confirms a prediction of a heuristic that says that coalescence times of pairs of non-crossing geodesics must have negative association. (joint with Firas Rassoul-Agha and Timo Seppäläinen)

Abstract:  We study the spatial covariance of the one-dimensional KPZ equation started from the flat initial profile, with the height function h(t,x) = log Z(t,x) given by the Cole-Hopf transform of the stochastic heat equation with constant initial data. At a fixed time t, we prove a sharp asymptotic for Cov(h(t,x), h(t,0)) as |x| tends to infinity: the covariance decays like a heat-kernel Gaussian envelope exp(-x^2/(4t)) times an explicit prefactor of order t^{3/2} |x|^{-2}. This is in sharp contrast with the narrow-wedge initial condition, where Gu and Pu recently proved a purely polynomial decay of order t/|x|. The far-field covariance of KPZ is therefore qualitatively, not just quantitatively, sensitive to the initial profile. The proof combines a Clark-Ocone representation, a boundary-layer reduction near the midpoint between the two points, and shear invariance and shear mixing of the white noise; along the way we obtain a closed-form second moment for the normalized continuum directed random polymer partition function.

In the second part we pass to two times. For the flat KPZ fixed point we prove quantitative two-time spatial decorrelation: for fixed times s and t, the covariance of the heights at (t,x) and (s,0) is bounded by C exp(-c|x|^3) for |x| at least 1. Unlike the fixed-time covariance, which is governed by the Airy_1 process, the two-time covariance involves the nonlinear variational evolution of the whole earlier profile; the proof combines cubic-exponential mixing of the Airy_1 process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, centered spatial averages normalized by the square root of the window size converge to a Gaussian process whose covariance is the space-integrated two-time correlation.

Based on joint work with Juan J. Jimenez (Auburn University) and with Fei Pu (Beijing Normal University).

References:

- L. Chen, J. J. Jiménez, Spatial covariance of KPZ from flat initial profile, arXiv:2603.14174.
- L. Chen, F. Pu, Two-time spatial decorrelation for the flat KPZ fixed point, arXiv:2607.17113.
- Y. Gu, F. Pu, Spatial decorrelation of KPZ from narrow wedge, arXiv:2506.23065.

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