In probability theory, the KPZ fixed point is a Markov field and conjectured to be a universal limit of a wide range of stochastic models forming the universality class of a non-linear stochastic partial differential equation called the KPZ equation. Even though the universality class was already introduced in 1986 with the KPZ equation itself, the KPZ fixed point was not concretely specified until 2021 when mathematicians Konstantin Matetski, Jeremy Quastel and Daniel Remenik gave an explicit description of the transition probabilities in terms of Fredholm determinants.
Introduction All models in the KPZ class have in common, that they have a fluctuating height function or some analogue function, that can be thought of as a function, that models the growth of the model by time. The KPZ equation itself is also a member of this class and the canonical model of modelling random interface growth. The strong KPZ universality conjecture conjectures that all models in the KPZ universality class converge under a specific scaling of the height function to the KPZ fixed point and only depend on the initial condition. Matetski-Quastel-Remenik constructed the KPZ fixed point for the ( 1 + 1 ) {\displaystyle (1+1)} -dimensional KPZ universality class (i.e. one space and one time dimension) on the polish space of upper semicontinous functions (UC) with the topology of local UC convergence. They did this by studying a particular model of the KPZ universality class the TASEP („Totally Asymmetric Simple Exclusion Process“) with general initial conditions and the random walk of its associated height function. They achieved this by rewriting the biorthogonal function of the correlation kernel, that appears in the Fredholm determinant formula for the multi-point distribution of the particles in the Weyl chamber. Then they showed convergence to the fixed point.
KPZ fixed point Let h ( t , x → ) {\displaystyle h(t,{\vec {x}})} denote a height function of some probabilistic model with ( t , x → ) ∈ R × R d {\displaystyle (t,{\vec {x}})\in \mathbb {R} \times \mathbb {R} ^{d}} denoting space-time. So far only the case for d = 1 {\displaystyle d=1} , also noted as ( 1 + 1 ) {\displaystyle (1+1)} , was deeply studied, therefore we fix this dimension for the rest of the article. In the KPZ universality class exist two equilibrium points or fixed points, the trivial Edwards-Wilkinson (EW) fixed point and the non-trivial KPZ fixed point. The KPZ equation connects them together. The KPZ fixed point is rather defined as a height function h ( t , x → ) {\displaystyle {\mathfrak {h}}(t,{\vec {x}})} and not as a particular model with a height function.
KPZ fixed point The KPZ fixed point ( h ( t , x ) ) t ≥ 0 , x ∈ R {\displaystyle ({\mathfrak {h}}(t,x))_{t\geq 0,x\in \mathbb {R} }} is a Markov process, such that the n-point distribution for x 1 < x 2 < ⋯ < x n ∈ R {\displaystyle x_{1}<x_{2}<\cdots <x_{n}\in \mathbb {R} } and t > 0 {\displaystyle t>0} can be represented as
P h ( 0 , ⋅ ) ( h ( t , x 1 ) ≤ a 1 , h ( t , x 2 ) ≤ a 2 , … , h ( t , x n ) ≤ a n ) = det ( I − K ) L 2 ( { x 1 , x 2 , … , x n } × R ) {\displaystyle \mathbb {P} _{{\mathfrak {h}}(0,\cdot )}({\mathfrak {h}}(t,x_{1})\leq a_{1},{\mathfrak {h}}(t,x_{2})\leq a_{2},\dots ,{\mathfrak {h}}(t,x_{n})\leq a_{n})=\det(I-K)_{L^{2}(\{x_{1},x_{2},\dots ,x_{n}\}\times \mathbb {R} )}}
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