In 1986\, Kardar\, Parisi and Zhang predicted t hat many planar random growth processes possess un iversal scaling behaviour. In particular\, models in the KPZ universality class have an analogue of the height function which is conjectured to conver ge 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 Virá \;g (2021) showed that the spatial increments of t he KPZ fixed point at any fixed time for general i nitial data are absolutely continuous with respect to Brownian motion on compacts.
\n\nIn this talk\, some recent work will be discussed that es tablishes the laws of spatial increments of the KP Z fixed point. These laws start from arbitrary ini tial data at any fixed time and exhibit quantitati ve comparison against rate two Brownian motion on compacts. The following functional relationship is obtained between the law of the spatial increment s of the KPZ fixed point\, &nu\; and the Wiener me asure:\n&nu\;(·\;) &le\; f(&mu\;(·\;)) \, for some explicit\, continuous strictly decreas ing function f vanishing at zero. This is a first step in the direction of establishing a con jecture by Hammond (2019) stating that the spatial increments of the KPZ fixed point have Radon-Niko dym derivative that is in L&infin\;-. This is based on joint work with Sourav Sark ar.
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