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Permutations with restricted movement

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

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  • 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.

    Mathematics Subject Classification: Primary: 37B50,; Secondary: 37B05, 37B40.


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  • Figure 2.1.  (a) The two-dimensional honeycomb lattice. (b) The fundamental domain

    Figure 2.2.  (a) Paths configuration corresponding to an elements in $ \Omega(G_{A_L}) $. (b) Paths configuration corresponding to an elements in $ \Omega(G_{A_+}) $

    Figure 2.3.  The graphs corresponding to $ A_+ $ and $ A_L $

    Figure 3.1.  The graph $ G_{A_L}' $

    Figure 3.2.  The quotient of $ L_{H} $ by the action of $ (2 {\mathbb Z})^2 $

    Figure 3.3.  A correspondence between a function in $ B_{3, 3}(A_L) $ and a perfect cover of $ \hat{V}_{3, 3} $ in $ G_{A_L} $

    Figure 3.4.  (a) The $ T $-gadget and the weights on its edges. (b) The construction of $ \hat{G} $ from $ G $

    Figure 3.5.  The correspondence between a restricted permutation of the honeycomb lattice, perfect matchings and permutations of $ {\mathbb Z}^2 $ restricted by $ A_L $

    Figure 3.6.  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) $

    Figure 4.1.  A locally admissible pattern which is not globally admissible where the restricting set is $ A_+ $

    Figure 4.2.  The extension of $ \pi_v $ to a $ \pi\in \Omega(A_\oplus) $

  • [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. CohnR. 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. KenyonA. 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. MarcusAn 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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