On the Manhattan pinball problem. (English) Zbl 1505.70037
Summary: We consider the periodic Manhattan lattice with alternating orientations going north-south and east-west. Place obstructions on vertices independently with probability \(0< p< 1\). A particle is moving on the edges with unit speed following the orientation of the lattice and it will turn only when encountering an obstruction. The problem is that for which value of \(p\) is the trajectory of the particle closed almost surely. We prove this is true for \(p> \frac{1}{2}-\varepsilon\) with some \(\varepsilon > 0\).
MSC:
| 70F99 | Dynamics of a system of particles, including celestial mechanics |
| 81Q99 | General mathematical topics and methods in quantum theory |
References:
| [1] | E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42 (1979), no. 10, 673-677. |
| [2] | M. Aizenman and G. Grimmett, Strict monotonicity for critical points in percolation and ferromagnetic models, J. Stat. Phys. 63 (1991), no. 5-6, 817-835. |
| [3] | E. J. Beamond, A. L. Owczarek, and J. Cardy, Quantum and classical localization and the Manhattan lattice, J. Phys. A 36 (2003), no. 41, 10251. · Zbl 1049.82047 |
| [4] | J. Bourgain and C. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389-426. · Zbl 1084.82005 |
| [5] | J. Cardy, Quantum network models and classical localization problems, Int. J. Mod. Phys. B 24 (2010), no. 12n13, 1989-2014. · Zbl 1195.82031 |
| [6] | R. Carmona, A. Klein, and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41-66. · Zbl 0615.60098 |
| [7] | J. Ding and C. Smart, Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice, Invent. Math. 219 (2020), no. 2, 467-506. · Zbl 1448.60148 |
| [8] | G. Grimmett, Percolation, 2nd ed., vol. 321, Springer Berlin Heidelberg, 1999. · Zbl 0926.60004 |
| [9] | G. Kozma and V. Sidoravicius, Lower bound for the escape probability in the Lorentz mirror model on the lattice, Isr. J. Math. 209 (2015), 683-685. · Zbl 1330.82012 |
| [10] | L. Li, Anderson-Bernoulli localization with large disorder on the 2D lattice, 2002.11580 (2020). |
| [11] | L. Li, Polynomial bound for the localization length of Lorentz mirror model on the 1D cylinder, 2010.05900 (2020). |
| [12] | T. Spencer, Duality, statistical mechanics and random matrices, Current Developments in Mathematics, International Press, Somerville, 2012, pp. 229-260 · Zbl 1292.82006 |
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