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Permutations with restricted movement
School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Israel |
A restricted permutation of a locally finite directed graph $ G = (V, E) $ is a vertex permutation $ \pi: V\to V $ for which $ (v, \pi(v))\in E $, for any vertex $ v\in V $. The set of such permutations, denoted by $ \Omega(G) $, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [
References:
[1] |
N. Chandgotia and T. Meyerovitch, Borel subsystems and ergodic universality for compact $\mathbb{Z}^d$-systems via specification and beyond, preprint, arXiv: 1903.05716. Google Scholar |
[2] |
H. Cohn, R. Kenyon and J. Propp,
A variational principle for domino tilings, J. Amer. Math. Soc., 14 (2001), 297-346.
doi: 10.1090/S0894-0347-00-00355-6. |
[3] |
M. Einsiedler and T. Ward, Ergodic Theory, Springer, 2013.
doi: 10.1007/978-0-85729-021-2. |
[4] |
D. Elimelech, Permutations with Restricted Movement, M.Sc Thesis, Ben-Gurion University, 2019, arXiv: 1911.02233. Google Scholar |
[5] |
M. E. Fisher,
Statistical mechanics of dimers on plane lattice, Phys. Rev., 124 (1961), 1664-1672.
doi: 10.1103/PhysRev.124.1664. |
[6] |
M. E. Fisher,
On the dimer solution of planar Ising models, J. of Math. Phys., 7 (1966), 1776-1781.
doi: 10.1063/1.1704825. |
[7] |
M. Hochman and T. Meyerovich,
A characterization of the entropies of multidimensional shifts of finite type, Annals of Math., 171 (2010), 2011-2038.
doi: 10.4007/annals.2010.171.2011. |
[8] |
P. W. Kasteleyn, The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225. Google Scholar |
[9] |
P. W. Kasteleyn,
Dimer statistics and phase transitions, Journal of Mathematical Physics, 4 (1963), 287-293.
doi: 10.1063/1.1703953. |
[10] |
R. Kenyon,
The planar dimer model with boundary: A survey, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13 (2000), 307-328.
|
[11] |
R. Kenyon, A. Okounkov and S. Sheffield,
Dimers and amoebae, Annals of Math., 163 (2006), 1019-1056.
doi: 10.4007/annals.2006.163.1019. |
[12] |
D. Kerr and H. Li,
Entropy and the variational principle for actions of sofic groups, Inventiones Mathematicae, 186 (2011), 501-558.
doi: 10.1007/s00222-011-0324-9. |
[13] |
D. Lind and B. H. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() |
[14] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Inventiones Mathematicae, 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[15] |
M. Misiurewicz,
A short proof of the variational principle for a $\mathbb{Z}^N$ action on a compact space, Asterisque, 40 (1976), 147-157.
|
[16] |
R. M. Robinson,
Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, 12 (1971), 177-209.
doi: 10.1007/BF01418780. |
[17] |
K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser/Springer Basel AG, Basel, 1995. |
[18] |
K. Schmidt and G. Strasser,
Permutations of $\mathbb{Z}^d$ with restricted movement, Studia Mathematica, 235 (2016), 137-170.
doi: 10.4064/sm8498-8-2016. |
[19] |
M. Schwartz and J. Bruck,
Constrained codes as networks of relations, IEEE Trans. Inform. Theory, 54 (2008), 2179-2195.
doi: 10.1109/TIT.2008.920245. |
[20] |
H. N. V. Temperley and M. E. Fisher,
Dimer problem in statistical mechanics–an exact result, Phil. Mag., 6 (1960), 1061-1063.
doi: 10.1080/14786436108243366. |
show all references
References:
[1] |
N. Chandgotia and T. Meyerovitch, Borel subsystems and ergodic universality for compact $\mathbb{Z}^d$-systems via specification and beyond, preprint, arXiv: 1903.05716. Google Scholar |
[2] |
H. Cohn, R. Kenyon and J. Propp,
A variational principle for domino tilings, J. Amer. Math. Soc., 14 (2001), 297-346.
doi: 10.1090/S0894-0347-00-00355-6. |
[3] |
M. Einsiedler and T. Ward, Ergodic Theory, Springer, 2013.
doi: 10.1007/978-0-85729-021-2. |
[4] |
D. Elimelech, Permutations with Restricted Movement, M.Sc Thesis, Ben-Gurion University, 2019, arXiv: 1911.02233. Google Scholar |
[5] |
M. E. Fisher,
Statistical mechanics of dimers on plane lattice, Phys. Rev., 124 (1961), 1664-1672.
doi: 10.1103/PhysRev.124.1664. |
[6] |
M. E. Fisher,
On the dimer solution of planar Ising models, J. of Math. Phys., 7 (1966), 1776-1781.
doi: 10.1063/1.1704825. |
[7] |
M. Hochman and T. Meyerovich,
A characterization of the entropies of multidimensional shifts of finite type, Annals of Math., 171 (2010), 2011-2038.
doi: 10.4007/annals.2010.171.2011. |
[8] |
P. W. Kasteleyn, The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225. Google Scholar |
[9] |
P. W. Kasteleyn,
Dimer statistics and phase transitions, Journal of Mathematical Physics, 4 (1963), 287-293.
doi: 10.1063/1.1703953. |
[10] |
R. Kenyon,
The planar dimer model with boundary: A survey, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13 (2000), 307-328.
|
[11] |
R. Kenyon, A. Okounkov and S. Sheffield,
Dimers and amoebae, Annals of Math., 163 (2006), 1019-1056.
doi: 10.4007/annals.2006.163.1019. |
[12] |
D. Kerr and H. Li,
Entropy and the variational principle for actions of sofic groups, Inventiones Mathematicae, 186 (2011), 501-558.
doi: 10.1007/s00222-011-0324-9. |
[13] |
D. Lind and B. H. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302.![]() ![]() |
[14] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Inventiones Mathematicae, 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[15] |
M. Misiurewicz,
A short proof of the variational principle for a $\mathbb{Z}^N$ action on a compact space, Asterisque, 40 (1976), 147-157.
|
[16] |
R. M. Robinson,
Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, 12 (1971), 177-209.
doi: 10.1007/BF01418780. |
[17] |
K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser/Springer Basel AG, Basel, 1995. |
[18] |
K. Schmidt and G. Strasser,
Permutations of $\mathbb{Z}^d$ with restricted movement, Studia Mathematica, 235 (2016), 137-170.
doi: 10.4064/sm8498-8-2016. |
[19] |
M. Schwartz and J. Bruck,
Constrained codes as networks of relations, IEEE Trans. Inform. Theory, 54 (2008), 2179-2195.
doi: 10.1109/TIT.2008.920245. |
[20] |
H. N. V. Temperley and M. E. Fisher,
Dimer problem in statistical mechanics–an exact result, Phil. Mag., 6 (1960), 1061-1063.
doi: 10.1080/14786436108243366. |








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