1. Big picture questions
We have organized these based on perceived level of difficulty within each category.-
Statistics of KPZ equation in $1+1$ dimension:
Problem 1.05.
Compute statistics for different initial data including $\mathcal{Z}_0(x)=1$ (flat) and $\mathcal{Z}_0(x)=e^{B(x)}$ for $B(x)$ a two-sided Brownian motion (equilibrium). -
Problem 1.1.
Compute multi-point (spatial) distribution for KPZ with various initial data. For example, with $\mathcal{Z}_0(x)=\delta_{x=0}$, let $$\mathcal{Z}(t,x)=e^{-\frac{x^2}{2t}+t^{1/3}A_t(t^{-2/3}x)-\frac{t}{24}}.$$ What is the distribution $F(\xi_1, \xi_2)$ such that $$\mathbb{P}\left(A_t(x_1)\leq\xi_1, A_t(x_2)\leq\xi_2\right)=F(\xi_1,\xi_2)?$$ Perhaps easier, show that as $t$ goes to infinity and space is scaled like $t^{2/3}x$, the process $A_{t}(t^{2/3}x)$ converges to the Airy$_2$ process in $x$. -
This should be much harder since the analogous distribution is not known in the setting of TASEP or LPP. It is not at all clear that there should be a formula.
Problem 1.15.
Compute multi-time distribution for KPZ. What is the distribution $F'_{t_1, t_2}(\xi_1, \xi_2)$ such that $$\mathbb{P}\left(A_{t_1}(x)\leq\xi_1, A_{t_2}(x)\leq\xi_2\right)=F'_{t_1, t_2}(\xi_1,\xi_2)?$$ -
Higher dimensions
Problem 1.2.
Make rigorous sense of the anisotropic KPZ equation $$\partial_t h=(\partial_x h)^2-(\partial_y h)^2+\Delta h+\xi$$ and show that the Gaussian free field is invariant. Is there a way of getting this equation out of the 2d Schur process dynamics of Borodin-Ferrari? Or perhaps out of the 2d q-Whittaker process dynamics of Borodin-Corwin? Is this related to $2d$ quantum Toda chain? -
Problem 1.25.
Prove that above 2 spatial dimensions, the KPZ equation is trivial (i.e., discretizations limit to the linearized equation, perhaps with a larger variance in the noise). -
Problem 1.3.
In 2 spatial dimensions is the measure valued solution of the stochastic heat equation related to the “exponential” of the dynamic GFF and hence quantum Louiville gravity? -
Problem 1.35.
Study polymer free energy and growth model fluctuation exponents in higher dimension? Compute limit shapes and fluctuation exponents. -
Structure at positive temperature / asymmetry:
Positive temperature polymers have triangular arrays associated with them (which are limits of the Macdonald processes). The marginal measure on a given level is a tropical analog of random matrix eigenvalue ensembles (such as GUE or LUE) which are determinantal point processes. These tropical point processes are no longer determinantal yet, there are still Fredholm determinants for Laplace type transforms of the analog of the largest or smallest eigenvalues. Information about all of the tropical eigenvalues should be accessible via Fredholm determinants (tropical analogs of gap probabilities for example).Problem 1.4.
What is the structure which replaces determinantal point processes and correlation functions and which allows for such computations? -
Associated to q-TASEP there is a triangular array (the q-Whittaker process). This whole array is helpful in computing things about q-TASEP.
Problem 1.45.
Is there such an array for ASEP or for some sort of transformed version of ASEP? -
The limit of the triangular array is the diffusion of O’Connell based on the quantum Toda lattice, or (after another limit) the KPZ$_T$ line ensemble. Both have Brownian Gibbs properties allowing paths to cross but at exponential cost (a soft analog of the non-intersecting Brownian Gibbs property for the Airy line ensemble.)The Karlin-McGregor formula underlies the solvability of zero temperature system since it writes non-intersecting line ensembles in terms of determinants.
Problem 1.5.
Is there an analog of the Karlin-McGregor formula when the non-intersecting conditioning is replaced by a softer form of conditioning? Does this explain the solvability and occurrence of Fredholm determinants? -
Expand the universality of the KPZ equation to:
Problem 1.55.
Growth model such as ballistic deposition. -
Problem 1.6.
Height functions associated to exclusion processes with longer range jumps, or environment dependent speed-changes. -
Problem 1.65.
Stochastic Hamilton-Jacobi equations $\partial_th=F(\nabla h)+\Delta h+\xi_\epsilon$ with $F$ scaled appropriately. -
Universality of the KPZ universality class:
Problem 1.75.
Prove the existence and uniqueness of the KPZ universality class fixed point introduced by Corwin-Quastel in 1 spatial dimension. That is to say, consider any growth process $h(t,x)$ and show that for an appropriate centering function $\overline{h}_{\epsilon}$, $$\lim_{\epsilon\rightarrow\infty}\epsilon^{1/2}h(\epsilon^{-3/2}t, \epsilon^{-1}x)-\overline{h}_\epsilon$$ has a limit as $\epsilon\to 0$ as a space-time process? Then show that the properties of this process identify it uniquely (see the conjectured properties in Corwin-Quastel). -
Problem 1.8.
Prove that the fixed point is attractive (i.e., universal in some class of models). -
Problem 1.85.
For example, prove universality of LPP and polymer (with respect to weight distributions). -
Problem 1.9.
Provide a variational explanation for the actual form of the GUE Tracy-Widom distribution in terms of the properties of the fixed point. Reduce the exact solvability to purely probabilistic terms.
Cite this as: AimPL: Kardar-Parisi-Zhang equation and universality class, available at http://aimpl.org/kpzuniversality.