# A KPZ Playbook

If you’ve heard my 2+1 KPZ talks before, or have just come to KITP, you might listen to first 10 mins here, then skip to the 55-min mark, where I wrap up the business of numerically-determined 2+1 flat & 3d radial KPZ universal limit distributions, higher-dimensional analogs of Tracy-Widom GOE & GUE, resp., as well as universal spatial covariance, analog of Airy-1, which allowed extraction of the KPZ nonlinearity lambda in the 2+1 kinetic roughening experiments of Palasantzas et al.
See,
PRL 109, 170602 (2012)
PRE 88, 042118 (2013)
EPL 105, 50001 (2014)
Available here

At the 1hr-mark (in the midst of Satya’s pointed question regarding the tails of these higher-d KPZ limit distributions…), I head to the finish-line. Because I was lined up to give the inaugural talk of this KITP KPZ workshop, Pierre asked me to discuss some “open questions”. So, here I provide 3 unsolved “Challenge Problems”; obviously, an eclectic, very personal list. Am hopeful that mathematicians at the workshop, armed with their impressive/powerful bag of tricks, can be of some help in this regard. Looking for analytical [rather than numerical…] solutions and, in one case, a mathematical bound. Each of the
3 Challenge Problems* comes with the same bounty- A bottle of Lagavulin, the traditional KPZ single-malt. Greg Huber, KITP Deputy Director (& inn-keeper), now has a bottle in hand. These are hard problems for sure, so require new/fresh mathematical viewpoints. In any case, as I promised Greg,
I’m good for the 3 bottles…

Grateful for any headway you folks can make; the first 2 problems, esp., have been weighing on me for some time now.

Best of luck to all for a productive, enjoyable workshop.
-TimHH

Bowie (1947-2016), RIP

*Prob #1: “Diamonds in the Rough…”
Analytical solution to Derrida-Griffiths [exact!] recursion relation for the limit distribution of the DPRM on a hierarchical lattice. Well-characterized numerically, the distribution is known exactly only in the zero & infinite-dimensional limits, where it is Gaussian(b=1) & Gumbel(b=\infty), respectively. This model is of great interest because, in contrast to Euclidean lattices, allows continuous interpolation between these dimensional extremes & provides strong (indirect) evidence that the KPZ/DPRM problem possesses no UCD. Analytical calculation of diamond DPRM
limit distribution (or, indeed, the fluctuation exponent omega) for any finite b.ne.1, integer or not, gets you the bottle...

The hierarchical DPRM is discussed in detail in my KPZ nano-review w/ Takeuchi; see Section 5 there.

Prob #2: “Mathematical Bound: 2+1 KPZ index” [of your choice….]
1+1 KPZ/DPRM/ASEP is Kyoto’s Golden Temple (elegant fractional exponents, exactly known: beta=1/3, alpha=1/2, z=3/2);
by contrast, 2+1 KPZ is gnarly & Godzilla-like. Kim & Kosterlitz propose beta=1/(d+1), which yields 1/4 in this dimension.
Best numerical estimates to-date:
beta=0.241(1)
- PRL 64, 1405 (1990), the long-surviving gold-standard, LeiHan working his magic; see follow-up paper Phys. Rev. A 45 7162 (1992)
- PRE 84, 061150 (2011); Kelling & Odor 2d Driven Dimers; Fig 3.
- PRE 88, 042118 (2013); sect. IV, here a multi-model estimate in the 2+1 KPZ stationary-state.

alpha_2+1=0.390+/-
Parisi & Pagnani- PRE 010101R (2015) quotes alpha=0.3869(4), though earlier estimate
Marinari, Pagnani & Parisi- JPA 33, 8181 (2000) had given alpha=0.393(3).
THH PRL 109, 170602 (2012), Table I gives, e.g., alpha= 0.388 for 2+1 KPZ Euler, and all values <0.390, via Krug-Meakin FSS method.

Note that Family-Vicsek scaling relation gives dynamic exponent z2+1=0.39/0.240=1.615 dangerously close to the Golden Ratio…
In this regard, note recent paper of Gunter Schutz & Co. on non equilibrium universality classes given by Kepler ratios (2/1, 3/2, 5/3, 8/5 etc..)
arising in the Fibonacci sequence. Obviously irrelevant since the PNAS paper focusses on 1d problem. Nevertheless, the authors address multilane scenario.
One therefore wonders about the many-lane limit, mentioned in passing in their Discussion section….

Prob #3: “DPRM/RSOS- Sierpinski Gasket” [or any other fractal substrate...]
The KPZ Renaissance commenced in 2010 w/ exact solutions & beautiful turbulent liqXtal expts for d=1 problem.
Exact solution for d=2 KPZ seems out of reach. What happens in between? Interesting (& perhaps not surprisingly…),
numerics indicate RSOS stochastic growth & DPRM problem on fractal support violates sacred KPZ identity alpha + z=2;
i.e., is not governed by KPZ eqn. See, e.g., JSTAT (2015) P08016.

Extra Credit: “Takeuchi Skewness Minimum”
A key signature manifest in the crossover from 1+1 KPZ transient (TW-GOE) to
stationary-state (Baik-Rains F0) limit distribution. This has been observed experimentally in the famous 1+1 KPZ turbulent liqXtal expts, and its universality well-documented numerically:
PRL 110, 210604 (2013)
PRE 89, 010103 (2014)
but needs to be teased out analytically by some deft, clever handiwork.