Invariant measure of gaps in degenerate competing three-particle systems

Abstract

We study the gap processes in a degenerate system of three particles interacting through their ranks. We obtain the Laplace transform of the invariant measure of these gaps, and an explicit expression for the corresponding invariant density. To derive these results, we start from the basic adjoint relationship characterizing the invariant measure, and apply a combination of two approaches: first, the invariance methodology of W. Tutte, thanks to which we compute the Laplace transform in closed form; second, a recursive compensation approach which leads to the density of the invariant measure as an infinite convolution of exponential functions. As in the case of Brownian motion with reflection or killing at the endpoints of an interval, certain Jacobi theta functions play a crucial role in our computations.

Figures (5)

• Figure 1: Left: graph of the function $\nu_2(u)$ for the parameters $(\lambda_1,\lambda_2)=(\frac{1}{6},\frac{5}{6})$. Right: graph of the function $\theta_{\mu}(q)$ in the case $\mu=\frac{1}{2}$.
• Figure 2: Example of graph of the density $\pi(u,v)$, on the left for the parameters $(\lambda_1,\lambda_2)=(\frac{1}{2},\frac{1}{2})$, on the right for $(\frac{1}{6},\frac{5}{6})$.
• Figure 3: The parabola $\mathcal{P}$ and its two branches $A_2^+$ and $A_2^-$, together with the lines of equation $2y-x=0$ and $2x-y=0$.
• Figure 4: The $y$-complex plane; the parabola $\mathcal{P}_2$ is drawn in blue; the domain $\mathcal{D}_2$ inside the parabola is represented by the blue dotted area; the domain $\{ y\in \mathbb{C}: \Re A_1^+(y)>0 \}$ is orange and is bounded by the curves of equation $\Re A_1^+(y)=0$; the curve $\Re A_1^-(y)=0$ is also represented by the orange dotted curve.
• Figure 5: The parabola $\mathcal{P}^*$ is represented in red, the line $\mathcal{L}_1$ in blue, $\mathcal{L}_2$ in green, the starting point $(a_0,b_0)$ in blue and $(a'_0,b'_0)$ in green. The automorphisms $\zeta$ and $\eta$ on the parabola allow us to define the sequences $(a_n,b_n)_{n\geqslant0}$ and $(a'_n,b'_n)_{n\geqslant0}$.

Theorems & Definitions (54)

• Theorem 1: Laplace transform, general case
• Corollary 1: Laplace transform, symmetric case
• Theorem 2: Density on the boundary, general case
• Corollary 2: Density on the boundary, symmetric case
• Theorem 3: Density of the invariant measure, general case
• Corollary 3: Density of the invariant measure, symmetric case
• Theorem 4
• Lemma 4: Analytic continuation
• proof
• Lemma 5: Parabola $\mathcal{P}_2$