
## 2. Open problems

1. ### Exact solvability

#### Problem 2.02.

Use the Macdonald processes formulas to prove the Baik-Ben Arous-Péché transition for the semi-discrete polymer with a finite number of non-zero drifts for the underlying Brownian motions (tuned critically). In the intermediate disorder scaling, derive the formula for the statistics of the stochastic heat equation started with $\mathcal{Z}_0(X) = {\bf 1}_{X\geq 0} Z^r(X)$ where $Z^N(X)$ is the partition function for the semi-discrete polymer with $r$ levels at time $X$ (the $r$ here represents the number of spikes in BBP and the drifts here of the $r$ Brownian motions can be generally chosen).
• ### Stochastic analysis

#### Problem 2.04.

Consider the multi-layer extension of the stochastic heat equation introduced by O’Connell and Warren. It is defined via chaos series – show that it can also be defined by spatially smoothing white-noise and then making sense of the Wick exponential in its formulation as a partition function for non-intersecting Brownian bridges in a space time white-noise environment.
• ### Exact solvability

#### Problem 2.06.

Motivated by success in solvable positive temperature polymers, try to study the polynuclear growth model with an underlying Brownian path measure – is there a Burkes theorem in this setting? Specifically, study the Poissonian last passage model but at positive temperature and with the Poisson points having random weights.
• ### Exact solvability

#### Problem 2.08.

Study the overlap for replicated paths of a polymer with respect to the quenched Gibbs measure. What is the distribution of the intersection local time for the continuum polymer? Is there replica symmetry or replica breaking in the strong disorder regime or intermediate disorder regime? Try to use the Macdonald processes contour integral formulas to compute more than just the expected overlap.
• ### Stochastic analysis

#### Problem 2.1.

Find a microscopic Gärtner transform (discrete Hopf-Cole transform) which turns the dynamics of $q$-TASEP into a discrete stochastic heat equation or better yet a polymer. Then prove this converges to the continuum stochastic heat equation.
• ### Interacting particle systems

#### Problem 2.12.

Bulk of $q$-Whittaker $2d$ dynamics. Invariant measure.
• ### Interacting particle systems

#### Problem 2.14.

Study $q$-TASEP hydrodynamics and other interacting particle systems properties such as second class particles, invariant distributions, and Burkes type theorem.
• ### Interacting particle systems

#### Problem 2.16.

What about $q$-ASEP, where particle can jump to right and left and feel a repulsion from their right or left neighbors: Is there Burkes Theorem? Do second class particle methods work?
• ### Gibbs line ensembles

#### Problem 2.18.

Consider a Brownian excursion on time interval $[-N,N]$ condition on fixed area $N$. Take $1/3, 2/3$ scalings around the edge. Do we get an “Airy-like” limiting fluctuation?
• ### Stochastic analysis

#### Problem 2.2.

Use the well-posedness theory for the KPZ equation to redo Bertini-Giacomin or to prove convergence of simpler discretizations of the KPZ equation such as the one previously studied by Sasamoto and Spohn.
• ### Polymer universality

#### Problem 2.22.

Consider intermediate scaling for polymer with $\beta=n^{-1/4}c(n)$ where $c(n)$ is any function which tends to infinity with $n$. Show universality of the $F_{GUE}$ fluctuations with a fluctuation exponent which may depend on $c(n)$.
1. Remark. This requires careful estimation of terms in the discrete chaos series which are further and further out as $n$ increases (reminiscent of Soshnikov’s approach to edge universality for Wigner matrices).
• ### Exact solvability

#### Problem 2.26.

Investigate the relationship between the tropical RSK correspondence and mirror symmetry: The entrance law for the Markov process which arises from adding columns under the tropical RSK correspondence arises via a critical point calculation.
1. Remark. The critical point has also arose in the study of mirror symmetry.
• ### Universality of initial data for (T)ASEP.

#### Problem 2.3.

Consider two initial conditions for (T)ASEP corresponding to height functions $h_1(x;t=0)$ and $h_2(x;t=0)$ which can be random, but are independent of each other. Hydrodynamic theory says that if for $i=1,2$, $\epsilon^{-1} h_i(\epsilon^{-1} x;t=0)\to \bar{h}(x;t=0)$ as $\epsilon\to 0$, then so does $\epsilon^{-1} h_i(\epsilon^{-1} x;t)\to \bar{h}(x;t)$ where $\bar{h}$ solves a Hamilton-Jacobi conservation law with quadratic flux. We would like a similar result, but for fluctuations. This result would show that if we assume for $i=1,2$, $$\epsilon^{1/2}\left[h_i(\epsilon^{-1}x;t=0)-\epsilon^{-3/2}\bar{h}(\epsilon^{1/2} x;0)\right]\to \tilde{h}(x;t=0)$$ as $\epsilon\to 0$, then so does $$\epsilon^{1/2}\left[h_i(\epsilon^{-1}x;\epsilon^{-3/2}t)-\epsilon^{-3/2}\bar{h}(\epsilon^{1/2}x;t)\right]\to \tilde{h}(x;t).$$
Just showing that $\tilde{h}_1=\tilde{h}_2$ is enough. In fact, $\tilde{h}(t,x)$ should be the result of evolving the initial fluctuations $\tilde{h}(x;t=0)$ according to the KPZ fixed point (random) semi-group. This might be easiest to show for TASEP by looking at the last passage percolation formulation and showing that modifying the initial height does not affect fluctuations in the KPZ scaling. This result is already known for the WASEP since then one can use Bertini-Giacomin’s approach to show a statement analogous to the above desired result.
• ### Universality of initial data for (T)ASEP.

#### Problem 2.32.

Show $T^{1/3}$ fluctuations for KPZ equation slightly out of equilibrium by using a variant of the second class particle method for WASEP near equilibrium.
• ### Gibbs line ensembles

#### Problem 2.34.

Show tightness and limiting Gibbs property for line ensembles besides the Airy line ensemble (e.g. Bessel process, Peary process, Sine process).
• ### Exact solvability

#### Problem 2.36.

Explain the occurrence of a boundary value problem (BVP) in the kernel for the continuum statistics for the Airy$_2$ process: $\mathbb{P}\left(A_2(\cdot)\leq g(\cdot)\right)=\det(I-M),$ where $M$ has a kernel given in terms of a BVP involving $g$. Is this due to the Karlin-McGregor formula for this infinite ensemble of lines? Study the analogous continuum statistics for the top curve of $N$ non-intersecting Brownian bridges or Brownian motions. This might explain where this BVP comes from.
• ### Exact solvability

#### Problem 2.38.

Study solvable variants of the log-Gamma polymer with different types of polymer paths.
• ### Polymer universality

#### Problem 2.4.

Determine limiting behavior of polymer without a sixth moment, and with $\beta\rightarrow0$ at the correct rate to have a limit.

Cite this as: AimPL: Kardar-Parisi-Zhang equation and universality class, available at http://aimpl.org/kpzuniversality.