Wall-crossing phenomenon for the liquid bin model

Abstract

We introduce the liquid bin model as a continuous-time deterministic dynamics, arising as the hydrodynamic limit of a discrete-time stochastic interacting particle system called the infinite bin model. For the liquid bin model, we prove the existence and uniqueness of a stationary evolution, to which the dynamics converges exponentially fast. The speed of the front of the system is explicitly computed as a continuous piecewise rational function of the parameters of the model, revealing an underlying wall-crossing phenomenon. We show that the regions on which the speed is rational are of non-empty interior and are naturally indexed by Dyck paths. We provide a complete description of the adjacency structure of these regions, which generalizes the Stanley lattice for Dyck paths. Finally we point out an intriguing connection to the topic of extensions of partial cyclic orders to total cyclic orders.

Figures (5)

• Figure 1: Illustration of the dynamics of the liquid bin model with parameters $N=2$, $a_1 = 1.5$, $p_1 =0.5$, $a_2 = 2.5$, $p_2 =1.5$. Cursor $1$ (respectively $2$) is represented in red (resp. blue): under it and to its right, there is always $a_1$ (resp. $a_2$) quantity of liquid. After a time $0.25$, starting from the configuration in the top-left, cursor $1$ goes from bin $1$ to bin $2$, yielding the configuration in the bottom-left. After an additional time $0.5$, cursor $2$ goes from bin $1$ to bin $2$ to obtain the configuration in the bottom-right. After waiting for an extra time $0.375$, the configuration in the top-right is reached. These configurations pertain to a stationary evolution with period $T=1.125$.
• Figure 2: On the left, the downward closed graph on vertex set $\llbracket1,5\rrbracket$ with edges $(1,2)$, $(1,3)$, $(2,3)$ and $(4,5)$. Each edge $(i,j)$ is such that $i<j$ and is directed from $i$ to $j$. We omit the depiction of the direction to avoid overloading the picture. On the right, the same DC graph, where we added an extra broken line for each vertex. The supremum of all the lines present in the picture is indicated in bold. It corresponds to the Dyck path of length $10$$+++---++--$, where $+$ (resp. $-$) denotes an "up" (resp. "down") vector.
• Figure 3: The Hasse diagram of the Stanley lattice for Dyck paths of length $6$. Each Dyck path stands at a node of the diagram. One Dyck path $P_1$ covers another Dyck path $P_2$ whenever $P_1$ lies above $P_2$ in the diagram and they are connected by an edge. Next to each edge is given a polynomial in the variables $(\underline d,\underline q)$. This polynomial vanishes on the boundary between the regions labeled by the two DC graphs that are in bijection with the two Dyck paths at each end of the edge. This polynomial is positive (resp. negative) on the region labeled by the graph corresponding to the Dyck path which is on the top (resp. bottom) end of the edge.
• Figure 4: Illustration of the coupling between the liquid bin model and the car model with parameters $N=3$, $d_1 = 1$, $d_2 =\frac{2}{3}$, $d_3 = 1$. On the left a picture of the bin configuration with unit-width bins. On the top right a picture of the same bin configuration with unit-height bins. On the bottom right the corresponding car configuration. For this bin configuration, we assume that the bins with indices greater than zero are empty so that the front position is $0$. The difference of positions between cars in the car model corresponds to the quantity of liquid in the bin model. The speed of each car is indicated above it.
• Figure 5: Illustration of the dynamics of the car model. The initial configuration is depicted at the top. On the figure, $v_i$ denotes the speed of the car with index $i$. After $0.5$ units of time, the car $-1$ arrives at position $a_1$. Therefore, the car $0$ starts moving at speed $q_1$. Then, after $0.75$ units of time, the car $-2$ arrives at position $a_2$. Therefore, the car $-1$ now moves at speed $q_2$.

Theorems & Definitions (70)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Example 1.6
• Proposition 1.7
• proof
• Conjecture 1.8
• Remark 2.1
• Remark 2.2