Simulations of stochastic higher spin vertex model

2015/03/12


The stochastic higher spin vertex model introduced in my paper with Ivan Corwin ([arXiv:1502.07374 [math.PR]][ivan6v]) generalizes the stochastic six vertex model considered by Borodin, Corwin, and Gorin, arXiv:1407.6729 [math.PR]. Here are some simulations related to this model.

Dear colleagues:

Feel free to use these pictures to illustrate your research in talks and papers, with attribution (CC BY-SA 4.0 (opens in new tab)). Some of the images are available in very high resolution upon request.

The stochastic higher spin vertex model generalizes a number of integrable stochastic particle systems on the line in the KPZ universality class:

Diagram showing how the stochastic higher spin vertex model generalizes integrable particle systems including TASEP and q-Hahn


The stochastic higher spin vertex model is probabilistic, which puts probability weights on configurations of arrows in some region. In our interpretation, the arrows will always point up and to the right. We will take this region to be the integer quadrant \(\mathbb{Z}_{\ge0}^2\).

The model has 4 parameters: \(q\), \(\nu\), \(\alpha\), and \(J\). The simplest case of the model arises when \(J=1\). If \(J\) is a positive integer, then there are at most \(J\) horizontal arrows allowed.

Model in \(J=1\) case

Let us first explain how \(J=1\) probability weights are assigned at a single vertex. The numbers of incoming arrows at a vertex (bottom and left) \(i_1\) and \(j_1\) are interpreted as an input which is fixed, and then the probability weight is put on numbers of outgoing arrows \(i_2\) and \(j_2\), such that \(i_1+j_1=i_2+j_2\):

Arrow configuration at a single vertex showing incoming arrows i1, j1 and outgoing arrows i2, j2 with conservation rule


For \(J=1\), the probability weights have the following form (where \(g=0,1,2,\ldots\)):

J=1 vertex probability weights expressed in terms of parameters q, nu, alpha, and g


If \(\nu=q^{-I}\) for \(I=1,2,\ldots\), then an additional restriction arises: there are at most \(I\) vertical arrows allowed.

Starting from any boundary conditions on \(\mathbb{Z}_{\ge0}^2\), we can recursively sample the configuration of arrows.

Simulations in \(J=1\) case

We choose the boundary conditions to consist of all incoming arrows from the left, and no incoming arrows from the bottom. In simulations below, the arrows are drawn without arrowheads, and thickness of an edge represents the number of arrows at that edge.

Stochastic six vertex model $(I=J=1)$

The limit shape and KPZ fluctuations theorems for the six vertex model were proven in arXiv:1407.6729 [math.PR].

Size $50$, $\alpha = -4$, $q = 1/3$, $\nu = 1/q$. Probability of an arrow to go straight up is $1/3$, and to go straight over is $2/3$.
Stochastic six-vertex model simulation, size 50, showing arrow paths on integer quadrant
Size $150$, other parameters same as above.
Stochastic six-vertex model simulation, size 150, showing limit shape formation
Size $500$, other parameters same as above.
Stochastic six-vertex model simulation, size 500, showing clear limit shape with KPZ fluctuations at boundary
Fluctuations of the height function of the above model (size $100$).
Height function fluctuations of the stochastic six-vertex model at size 100

$I=4$, $J=1$

Size $350$, $\alpha = -18$, $q = 0.5$, $\nu = 1/q^4$.
Higher spin vertex model with I=4, J=1, size 350, showing multiple arrow layers
A zoom into the above picture.
Zoomed view of the I=4, J=1 higher spin vertex model showing arrow detail

General \(\nu\) and \(J=1\)

Size $50$, $\alpha = 0.8$, $q = 0.5$, $\nu = 0.4$. If there are more than 10 vertical arrows at an edge, the boldness is still like that for 10 arrows.
General nu vertex model, size 50, with unbounded vertical arrows and edge thickness proportional to arrow count
Size $400$, other parameters same as above.
General nu vertex model, size 400, showing large-scale limit shape with packed region
A zoom into the above picture.
Zoomed view of the general nu vertex model at size 400

Model in general \(J\) case

For an integer \(J\), the vertex weights may be obtained by a fusion procedure (first performed by Kirillov and Reshetikhin; J. Phys. A, 1987), by putting \(J=1\) layers with changing parameters: \(\alpha, q\alpha, q^2\alpha, \ldots, q^{J-1}\alpha\). For example, for \(J=2\) we obtain the following vertex weights:

J=2 fused vertex probability weights obtained from the Kirillov-Reshetikhin fusion procedure


In general, the vertex weights can be expressed through the basic hypergeometric functions, and, moreover, are always polynomials in \(q^J\). Therefore, one can perform an analytic continuation in \(q^J\), and regard \(J\) (or rather \(q^J\)) as a complex parameter. However, for \(J\) not integer, resulting systems in general fail to become stochastic. There is at least one stochastic system which can be obtained via this analytic continuation, it arises at \(\alpha=-\nu\), \(q^J\alpha=-\mu\), and it is the \(q\)-Hahn system introduced by Povolotsky in arXiv:1308.3250 [math-ph].

Simulations in general \(J\) case

We choose the boundary conditions to consist of \(J\) incoming arrows from the left, and no incoming arrows from the bottom. In simulations below, the arrows are drawn without arrowheads, and thickness of an edge represents the number of arrows at that edge.

Size around $270$, $\alpha = -18$, $q = .5$, $\nu = 1/q^4$, $J=3$. At most 4 vertical arrows and at most 3 horizontal arrows are allowed, therefore there is a packed region on the left.
Higher spin vertex model with I=4, J=3, size 270, showing packed region on the left from the horizontal arrow constraint
A zoom into the above picture.
Zoomed view of the I=4, J=3 higher spin vertex model showing arrow detail near the packed region boundary
Fluctuations of the stochastic six vertex model
Fluctuations of the stochastic six vertex model