Scaling law for membrane lifetime

TL;DR

This work analyzes membrane lifetimes in the BFSS matrix model by studying chaotic decay in two reduced settings: a four-hill toy model and a truncated membrane sector of BFSS. Lifetimes are extracted from escape/decay-time histograms and shown to follow simple power-law scalings with energy $E$, coupling $g$, and cutoff scales ($\sigma$ or $l$), with exponents determined numerically and found to be robust to initial conditions. The extracted exponents are around $\alpha \approx -0.62$ to $-0.65$ and confirm a $1/2$-power dependence on the mass, indicating universality beyond naive dimensional analysis. These results connect chaotic scattering, membrane dynamics, and potential real-time evaporation processes of matrix black holes, motivating analytic proofs and further explorations of links to Lyapunov exponents and fractal time-delay structures.

Abstract

Membrane configurations in the Banks-Fischler-Shenker-Susskind matrix model are unstable due to the existence of flat directions in the potential and the decay process can be seen as a realization of chaotic scattering. In this note, we compute the lifetime of a membrane in a reduced model. The resulting lifetime exhibits scaling laws with respect to energy, coupling constant and a cut-off scale. We numerically evaluate the scaling exponents, which cannot be fixed by the dimensional analysis. Finally, some applications of the results are discussed.

Figures (6)

• Figure 1: The shape of the potential with $g=1$ and $\sigma=1$ .
• Figure 2: A histogram for event number $n(T)$ vs. escape time $T$ is drawn (left). The log of event number is fitted by a linear function (right). The parameter values are $E=0.1\,,~g=1\,,~\sigma=1$ and $m=1$ .
• Figure 3: Log-log plots of lifetime $\tau$ against one of $\{ E, g, \sigma, m \}$ with the other parameters fixed.
• Figure 4: The shape of the potential with $g=1$.
• Figure 5: A histogram for event number $n(\tilde{T})$ vs. decay time $\tilde{T}$ is drawn (left). The log of event number is fitted by a linear function (right). The parameter values are $E=0.1\,,~g=1\,,~l=5$ and $m=1$ .