<p>In 1986, Kardar, Parisi and Zhang predicted that many planar random growth processes possess universal scaling behaviour. In particular, models in the KPZ universality class have an analogue of the height function which is conjectured to converge at large time and small length scales under the KPZ 1:2:3 scaling to a universal Markov process, called the KPZ fixed point. Sarkar and Virág (2021) showed that the spatial increments of the KPZ fixed point at any fixed time for general initial data are absolutely continuous with respect to Brownian motion on compacts.</p>

<p>In this talk, some recent work will be discussed that establishes the laws of spatial increments of the KPZ fixed point. These laws start from arbitrary initial data at any fixed time and exhibit quantitative comparison against rate two Brownian motion on compacts. The following functional relationship is obtained between the law of the spatial increments of the KPZ fixed point, ν and the Wiener measure:
ν(·) ≤ f(μ(·)), for some explicit, continuous strictly decreasing function <i>f</i> vanishing at zero. This is a first step in the direction of establishing a conjecture by Hammond (2019) stating that the spatial increments of the KPZ fixed point have Radon-Nikodym derivative that is in <i>L</i>^∞-. This is based on joint work with Sourav Sarkar.</p>