Extending the multipoint distribution formula of the KPZ fixed point to arbitrary (non-compactly-supported) upper semicontinuous initial conditions

The KPZ fixed point is the universal space-time scaling limit of the Kardar–Parisi–Zhang (KPZ) universality class. Liao and Liu (arXiv:2509.03246, 2025) proved the first explicit formula for the joint distribution of the fixed point at arbitrarily many space-time points — a nested contour integral of a Fredholm determinant $\mathbf D_{\mathfrak h}$ — valid for initial conditions $\mathfrak h$ that are upper semicontinuous and compactly supported. The dependence on the initial condition enters through a single characteristic kernel $\chi_{\mathfrak h}(\eta,\xi)$ defined by Brownian hitting expectations, and the authors explicitly leave open the extension to arbitrary upper semicontinuous initial conditions (flat, Brownian, stationary, and general non-compact profiles), where $\chi_{\mathfrak h}$ may fail to be a pointwise-defined function — for the flat initial condition one can show that it degenerates to a Dirac delta kernel — and may require a distributional or operator-theoretic interpretation. This problem asks for such an interpretation and a proof of the multipoint formula for all $\mathfrak h$ in the full space $\mathrm{UC}$. Known follow-up work constructs the periodic analogue (the periodic KPZ fixed point with general initial conditions) and related determinant identities, but does not address the extension on the line; the direct equivalence of the distinct narrow-wedge multipoint formulas also remains unproved.

Background

The Kardar–Parisi–Zhang (KPZ) universality class consists of $1+1$-dimensional random growth models — interacting particle systems, directed polymers, stochastic PDEs — whose height functions $H(x,t)$ share universal exponents: fluctuations of order $t^{1/3}$, correlated over distances $t^{2/3}$, so that under the $1:2:3$ scaling $x\mapsto\varepsilon^{-1}x$, $t\mapsto\varepsilon^{-3/2}t$, $H\mapsto\varepsilon^{-1/2}H$ the rescaled fields have non-trivial limits. The conjectural universal limit is the KPZ fixed point $\mathcal H(\mathsf x,\tau;\mathfrak h)$, a Markov process on the space of upper semicontinuous functions on $\mathbb R$ (the natural state space for height profiles), constructed by Matetski, Quastel and Remenik as the scaling limit of the totally asymmetric simple exclusion process (TASEP — a particle system on $\mathbb Z$ in which each particle jumps one site to the right at rate $1$ provided the target site is empty; its height function encodes the integrated particle current). They computed its transition probabilities explicitly as Fredholm determinants (the determinant $\det(I+K)$ of the identity plus a trace-class integral operator) and obtained, for arbitrary initial conditions $\mathfrak h$, the joint law of the height at finitely many points at a fixed time. The fixed point also has a variational (Hopf–Lax) representation driven by the directed landscape, the universal random energy landscape of the class.

The genuinely dynamical question — the joint law of $\mathcal H(\mathsf x_1,\tau_1;\mathfrak h),\ldots,\mathcal H(\mathsf x_m,\tau_m;\mathfrak h)$ at finitely many space-time points (the 'multipoint' or 'multi-time' distributions) — is far less understood, because the determinantal-point-process structure behind the equal-time formulas is lost in the time direction. Until the work of Liao and Liu, explicit multipoint formulas existed only for special initial data: the narrow wedge (droplet) initial condition $\mathfrak h=-\infty\mathbf 1_{\mathsf x\neq 0}$, via discrete polynuclear growth (Johansson–Rahman) and via TASEP (Z. Liu, who also treated the flat initial condition $\mathfrak h\equiv 0$); two-point distributions for special non-droplet geometries were obtained later (Johansson–Rahman; Rahman).

Liao and Liu proved the first multipoint formula for general compactly supported initial conditions. Write $\mathrm{UC}$ for the space of upper semicontinuous $\mathfrak h:\mathbb R\to[-\infty,\infty)$, $\mathfrak h\not\equiv-\infty$, with $\limsup_{\mathsf x\to\pm\infty}\mathfrak h(\mathsf x)/\mathsf x^2\le 0$ — the largest class of initial conditions for which the fixed point is almost surely finite at positive times — and $\mathrm{UC}_c\subset\mathrm{UC}$ for the dense subspace of profiles equal to $-\infty$ outside a compact set. For $\mathfrak h\in\mathrm{UC}_c$ and ordered space-time points $(\mathsf x_1,\tau_1)\prec\cdots\prec(\mathsf x_m,\tau_m)$ (ordered by time, then space), they showed that the joint distribution function equals a nested contour integral $\oint\cdots\oint\,\mathbf D_{\mathfrak h}(z_1,\ldots,z_{m-1})$ of a Fredholm determinant whose kernel acts on nested contours in the complex plane, with all dependence on the initial condition concentrated in a single entry, the characteristic kernel $\chi_{\mathfrak h}(\eta,\xi)$. This kernel is defined by hitting expectations of a two-sided Brownian motion (diffusivity $2$) to the hypograph of $\mathfrak h$ (the region on or below the graph) — a probabilistic representation adapted from the Brownian hitting operators used by Matetski–Quastel–Remenik for the transition probabilities. For the narrow wedge it evaluates to $1/(\eta-\xi)$, recovering the earlier TASEP-based formula, and when all times coincide the formula is shown to reduce to the path-integral (iterated transition kernel) formula of Matetski–Quastel–Remenik.

The limitation is the compactness assumption, and the authors stress that it is structural rather than technical: their bound on the growth of $\chi_{\mathfrak h}$, on which the convergence of the Fredholm construction relies, uses that the Brownian hitting times of the hypograph are bounded by the size of the support. For non-compactly-supported profiles — the flat initial condition $\mathfrak h\equiv 0$, Brownian initial data, stationary initial data — the kernel $\chi_{\mathfrak h}(\eta,\xi)$ need not be pointwise well-defined; already for the flat initial condition one can show it degenerates to the Dirac delta kernel $\delta_{\eta=-\xi}$, a genuine distribution rather than a function. The authors leave the extension to all $\mathfrak h\in\mathrm{UC}$ explicitly as a future project and suggest two possible routes: interpreting $\chi_{\mathfrak h}$ case by case as a distributional kernel, or conjugating the contour formulation to operators on real-variable function spaces so that limits of expanding supports exist at the operator level. Neither route has been carried out.

Problem Statement

This problem is the future project explicitly stated by Liao and Liu (arXiv:2509.03246): extend their multipoint distribution formula for the KPZ fixed point from compactly supported initial conditions $\mathfrak h\in\mathrm{UC}_c$ to arbitrary upper semicontinuous initial conditions $\mathfrak h\in\mathrm{UC}$. Concretely:

(1) Analytic meaning of the characteristic kernel. Give the kernel $\chi_{\mathfrak h}(\eta,\xi)$ — defined for compactly supported $\mathfrak h$ by the Brownian hitting formula (Definition 2.2 of the source) — a rigorous meaning for non-compactly-supported $\mathfrak h$, for instance as a pointwise function on the integration contours, a distribution (for example a tempered distribution), or a bounded operator on a suitable real-variable function space, and characterize the growth-at-infinity conditions on $\mathfrak h$ under which this meaning exists and is compatible with the Fredholm determinant $\mathbf D_{\mathfrak h}$. The flat initial condition $\mathfrak h\equiv 0$ — for which one can show from the source's formula that the kernel degenerates to the Dirac delta kernel $\delta_{\eta=-\xi}$ — together with the Brownian and stationary initial conditions, are the central test cases. The characterization serves the extension goal of item (2): the source states the target as extending the formula to any $\mathfrak h\in\mathrm{UC}$, so an interpretation valid only on a proper subclass of $\mathrm{UC}$ would constitute partial progress rather than a solution of the stated problem.

(2) The extended formula. Prove that for all $\mathfrak h\in\mathrm{UC}$, all finite ordered sets of space-time points $(\mathsf x_1,\tau_1)\prec\cdots\prec(\mathsf x_m,\tau_m)$ in $\mathbb R\times\mathbb R_+$, and all thresholds $\beta_1,\ldots,\beta_m\in\mathbb R$, the multipoint distribution of the KPZ fixed point is given in the extended framework by the nested contour integral of $\mathbf D_{\mathfrak h}$:

$$\mathbb{P}\Big(\bigcap_{\ell=1}^m\bigl\{\mathcal H(\mathsf x_\ell,\tau_\ell;\mathfrak h)\le\beta_\ell\bigr\}\Big)=\oint_0\frac{\mathrm d z_1}{2\pi\mathrm i\,z_1(1-z_1)}\cdots\oint_0\frac{\mathrm d z_{m-1}}{2\pi\mathrm i\,z_{m-1}(1-z_{m-1})}\,\mathbf D_{\mathfrak h}(z_1,\ldots,z_{m-1}).$$

Here each $\oint_0$ denotes integration over a small positively oriented contour encircling the origin, and $\mathbf D_{\mathfrak h}$ is the Fredholm determinant of the source paper, acting on nested contours in the complex plane, whose dependence on the initial condition $\mathfrak h$ enters only through the single characteristic kernel $\chi_{\mathfrak h}(\eta,\xi)$. The authors suggest two possible routes — defining $\chi_{\mathfrak h}$ case by case as a distributional kernel, or conjugating the contour formulation to operators on real-variable spaces so that the limits of expanding supports exist in operator norm — but any rigorous route is admissible.

The source paper additionally records two adjacent open directions that are not part of the present problem: a direct rigorous proof of the equivalence between the distinct narrow-wedge multipoint formulas of Johansson–Rahman (arXiv:1906.01053) and Z. Liu (arXiv:1907.09876), and the derivation of multipoint formulas directly from microscopic models other than TASEP.

References

1. Y. Liao and Z. Liu, "Multipoint distributions of the KPZ fixed point with compactly supported initial conditions", arXiv:2509.03246 (2025), DOI 10.48550/arXiv.2509.03246, https://arxiv.org/abs/2509.03246.
2. K. Matetski, J. Quastel, and D. Remenik, "The KPZ fixed point", Acta Mathematica 227(1) (2021) 115–203, DOI 10.4310/acta.2021.v227.n1.a3; arXiv:1701.00018, https://arxiv.org/abs/1701.00018.
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