In 1986\, Kardar\, Parisi and Zhang predicted that many pla nar random growth processes possess universal scaling behaviour. In partic ular\, 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 Virá\;g (2021) showed that the spati al increments of the KPZ fixed point at any fixed time for general initial data are absolutely continuous with respect to Brownian motion on compact s.
\n\nIn this talk\, some recent work will be discussed that establ ishes the laws of spatial increments of the KPZ fixed point. These laws st art from arbitrary initial data at any fixed time and exhibit quantitative comparison against rate two Brownian motion on compacts. The following fu nctional relationship is obtained between the law of the spatial increment s of the KPZ fixed point\, &nu\; and the Wiener measure:\n&nu\;(·\;) &le\; f(&mu\;(·\;))\, for some explicit\, continuous strictly decre asing function f vanishing at zero. This is a first step in the dir ection of establishing a conjecture by Hammond (2019) stating that the spa tial increments of the KPZ fixed point have Radon-Nikodym derivative that is in L&infin\;-. This is based on joint work with Soura v Sarkar.
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