# American Institute of Mathematical Sciences

September  2021, 41(9): 4319-4349. doi: 10.3934/dcds.2021038

## Permutations with restricted movement

 School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Israel

Received  January 2020 Revised  October 2020 Published  September 2021 Early access  March 2021

Fund Project: This work was supported by the Israel Science Foundation (ISF) under grant No. 270/18 and 1052/18

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 [18] of restricted ${\mathbb Z}^d$ permutations, in which $\Omega(G)$ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted ${\mathbb Z}^d$-permutations. We discuss the global and local admissibility of patterns, in the context of restricted ${\mathbb Z}^d$-permutations. Finally, we review the related models of injective and surjective restricted functions.

Citation: Dor Elimelech. Permutations with restricted movement. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4319-4349. doi: 10.3934/dcds.2021038
##### 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. [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. [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. [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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##### 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. [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. [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. [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.
(a) The two-dimensional honeycomb lattice. (b) The fundamental domain
(a) Paths configuration corresponding to an elements in $\Omega(G_{A_L})$. (b) Paths configuration corresponding to an elements in $\Omega(G_{A_+})$
The graphs corresponding to $A_+$ and $A_L$
The graph $G_{A_L}'$
The quotient of $L_{H}$ by the action of $(2 {\mathbb Z})^2$
A correspondence between a function in $B_{3, 3}(A_L)$ and a perfect cover of $\hat{V}_{3, 3}$ in $G_{A_L}$
(a) The $T$-gadget and the weights on its edges. (b) The construction of $\hat{G}$ from $G$
The correspondence between a restricted permutation of the honeycomb lattice, perfect matchings and permutations of ${\mathbb Z}^2$ restricted by $A_L$
The corresponding graph for $G_{A_+}$ from Theorem 1. Black points represent vertices of the form $(I, n)$ and white points represent vertices of the form $(O, n)$
A locally admissible pattern which is not globally admissible where the restricting set is $A_+$
The extension of $\pi_v$ to a $\pi\in \Omega(A_\oplus)$
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