The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that last passage geodesics converge to random functions with Hölder-2/3 − continuous paths. This work completes the construction of the central object in the Kardar-Parisi-Zhang universality class, the directed landscape.

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We show that classical integrable models of last passage percolation and the related nonintersecting random walks converge uniformly on compact sets to the Airy line ensemble. Our core approach is to show convergence of nonintersecting Bernoulli random walks in all feasible directions in the parameter space. We then use coupling arguments to extend convergence to other models.

The parabolic Airy line ensemble A is a central limit object in the KPZ universality class and related areas. On any compact set K = {1, . . . , k} × [a, a + t], the law of the recentered ensemble A−A(a) has a density X K with respect to the law of k independent Brownian motions. We show thatwhere S is an explicit, tractable, non-negative function of f . We use this formula to show that X K is bounded above by a K-dependent constant, give a sharp estimate on the size of the set where X K < ǫ as ǫ → 0, and prove a large deviation principle for A. We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two-point tail bounds that are stronger than those for Brownian motion. The paper is essentially self-contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.1. Almost surely, L satisfies L i (r) > L i+1 (r) for all pairs (i, r) ∉ 1, k × (0, t).

A sorting network is a geodesic path from 12 · · · n to n · · · 21 in the Cayley graph of S n generated by adjacent transpositions. For a uniformly random sorting network, we establish the existence of a local limit of the process of space-time locations of transpositions in a neighbourhood of an for a ∈ [0, 1] as n → ∞. Here time is scaled by a factor of 1/n and space is not scaled.The limit is a swap process U on Z. We show that U is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on a is through time scaling by a factor of a(1 − a).To establish the existence of U , we find a local limit for staircase-shaped Young tableaux. These Young tableaux are related to sorting networks through a bijection of Edelman and Greene.