Percolation games, probabilistic cellular automata, and the hard-core model View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2018-10-26

AUTHORS

Alexander E. Holroyd, Irène Marcovici, James B. Martin

ABSTRACT

Let each site of the square lattice Z2 be independently assigned one of three states: a trap with probability p, a target with probability q, and open with probability 1-p-q, where 0a move consists of moving the token from its current site x to either x+(0,1) or x+(1,0). A player who moves the token to a trap loses the game immediately, while a player who moves the token to a target wins the game immediately. Is there positive probability that the game is drawn with best play—i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain family of elementary one-dimensional probabilistic cellular automata (PCA). These automata have been studied in the contexts of enumeration of directed lattice animals, the golden-mean subshift, and the hard-core model, and their ergodicity has been noted as an open problem by several authors. We prove that these PCA are ergodic, and correspondingly that the game on Z2 has no draws. On the other hand, we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions, including an oriented version of the even sublattice of Zd in all d≥3. This is proved via a dimension reduction to a hard-core lattice gas in dimension d-1. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice Zd for d≥3, but here our method encounters a fundamental obstacle. More... »

PAGES

1-31

References to SciGraph publications

  • 1975-02. Phase transitions of hard sphere lattice gases in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2001-04. Reliable Cellular Automata with Self-Organization in JOURNAL OF STATISTICAL PHYSICS
  • 1974-06. The use of reflection as symmetry operation in connection with Peierls' argument in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1972-12. An ergodic theorem for interacting systems with attractive interactions in PROBABILITY THEORY AND RELATED FIELDS
  • 1982-09. The positive rates problem for attractive nearest neighbor spin systems on ℤ in PROBABILITY THEORY AND RELATED FIELDS
  • 2008. A Connection between ℤn and Generalized Triangular Grids in ADVANCES IN VISUAL COMPUTING
  • 2001-04. A Reader's Guide to Gacs's “Positive Rates” Paper in JOURNAL OF STATISTICAL PHYSICS
  • 1978. Bernoulli and Markov stationary measures in discrete local interactions in LOCALLY INTERACTING SYSTEMS AND THEIR APPLICATION IN BIOLOGY
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00440-018-0881-6

    DOI

    http://dx.doi.org/10.1007/s00440-018-0881-6

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1107859896


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0104", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Statistics", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Seattle University", 
              "id": "https://www.grid.ac/institutes/grid.263306.2", 
              "name": [
                "Seattle, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Holroyd", 
            "givenName": "Alexander E.", 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Institut \u00c9lie Cartan de Lorraine", 
              "id": "https://www.grid.ac/institutes/grid.462063.5", 
              "name": [
                "Institut Elie Cartan de Lorraine, Universit\u00e9 de Lorraine, Campus Scientifique, BP 239, 54506, Vandoeuvre-l\u00e8s-Nancy Cedex, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Marcovici", 
            "givenName": "Ir\u00e8ne", 
            "id": "sg:person.011242605267.32", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011242605267.32"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "University of Oxford", 
              "id": "https://www.grid.ac/institutes/grid.4991.5", 
              "name": [
                "Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, Oxford, UK"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Martin", 
            "givenName": "James B.", 
            "id": "sg:person.0745225500.62", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0745225500.62"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf00539839", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003004744", 
              "https://doi.org/10.1007/bf00539839"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/rsa.20281", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1004532402"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/rsa.20281", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1004532402"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00679137", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1011876319", 
              "https://doi.org/10.1007/bf00679137"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00679137", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1011876319", 
              "https://doi.org/10.1007/bf00679137"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00679137", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1011876319", 
              "https://doi.org/10.1007/bf00679137"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/s0012-365x(97)00109-x", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1023290641"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1023/a:1004823720305", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1031444084", 
              "https://doi.org/10.1023/a:1004823720305"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bfb0070087", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1033207441", 
              "https://doi.org/10.1007/bfb0070087"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.tcs.2014.09.009", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1036268022"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01614095", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040866374", 
              "https://doi.org/10.1007/bf01614095"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01614095", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040866374", 
              "https://doi.org/10.1007/bf01614095"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01646324", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041031628", 
              "https://doi.org/10.1007/bf01646324"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01646324", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041031628", 
              "https://doi.org/10.1007/bf01646324"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-540-89646-3_115", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1044187555", 
              "https://doi.org/10.1007/978-3-540-89646-3_115"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-540-89646-3_115", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1044187555", 
              "https://doi.org/10.1007/978-3-540-89646-3_115"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/0001-8708(79)90040-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1045595030"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1023/a:1004824203467", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1049184356", 
              "https://doi.org/10.1023/a:1004824203467"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1017/s0963548303006035", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1050562815"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevlett.51.853", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060789596"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1103/physrevlett.51.853", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1060789596"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/1110026", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062865876"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1214/12-aihp530", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1064392840"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1214/14-aop918", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1064394516"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1214/16-aap1190", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1064395532"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1214/aoap/1177004698", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1064397888"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1239/aap/1386857853", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1064441128"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1142/s0129054117500174", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1085554926"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2018-10-26", 
        "datePublishedReg": "2018-10-26", 
        "description": "Let each site of the square lattice Z2 be independently assigned one of three states: a trap with probability p, a target with probability q, and open with probability 1-p-q, where 0
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00440-018-0881-6'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00440-018-0881-6'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00440-018-0881-6'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00440-018-0881-6'


     

    This table displays all metadata directly associated to this object as RDF triples.

    147 TRIPLES      21 PREDICATES      45 URIs      16 LITERALS      5 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00440-018-0881-6 schema:about anzsrc-for:01
    2 anzsrc-for:0104
    3 schema:author Nc4f3d38977794620ac683943a36eaabb
    4 schema:citation sg:pub.10.1007/978-3-540-89646-3_115
    5 sg:pub.10.1007/bf00539839
    6 sg:pub.10.1007/bf00679137
    7 sg:pub.10.1007/bf01614095
    8 sg:pub.10.1007/bf01646324
    9 sg:pub.10.1007/bfb0070087
    10 sg:pub.10.1023/a:1004823720305
    11 sg:pub.10.1023/a:1004824203467
    12 https://doi.org/10.1002/rsa.20281
    13 https://doi.org/10.1016/0001-8708(79)90040-9
    14 https://doi.org/10.1016/j.tcs.2014.09.009
    15 https://doi.org/10.1016/s0012-365x(97)00109-x
    16 https://doi.org/10.1017/s0963548303006035
    17 https://doi.org/10.1103/physrevlett.51.853
    18 https://doi.org/10.1137/1110026
    19 https://doi.org/10.1142/s0129054117500174
    20 https://doi.org/10.1214/12-aihp530
    21 https://doi.org/10.1214/14-aop918
    22 https://doi.org/10.1214/16-aap1190
    23 https://doi.org/10.1214/aoap/1177004698
    24 https://doi.org/10.1239/aap/1386857853
    25 schema:datePublished 2018-10-26
    26 schema:datePublishedReg 2018-10-26
    27 schema:description Let each site of the square lattice Z2 be independently assigned one of three states: a trap with probability p, a target with probability q, and open with probability 1-p-q, where 0<p+q<1. Consider the following game: a token starts at the origin, and two players take turns to move, where a move consists of moving the token from its current site x to either x+(0,1) or x+(1,0). A player who moves the token to a trap loses the game immediately, while a player who moves the token to a target wins the game immediately. Is there positive probability that the game is drawn with best play—i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain family of elementary one-dimensional probabilistic cellular automata (PCA). These automata have been studied in the contexts of enumeration of directed lattice animals, the golden-mean subshift, and the hard-core model, and their ergodicity has been noted as an open problem by several authors. We prove that these PCA are ergodic, and correspondingly that the game on Z2 has no draws. On the other hand, we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions, including an oriented version of the even sublattice of Zd in all d≥3. This is proved via a dimension reduction to a hard-core lattice gas in dimension d-1. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice Zd for d≥3, but here our method encounters a fundamental obstacle.
    28 schema:genre research_article
    29 schema:inLanguage en
    30 schema:isAccessibleForFree true
    31 schema:isPartOf sg:journal.1053886
    32 schema:name Percolation games, probabilistic cellular automata, and the hard-core model
    33 schema:pagination 1-31
    34 schema:productId N24e3425afcaf4038a61329c6a75df637
    35 N333b2216adb94dccbd53c0d8c4fb6338
    36 Nc115fb0ca0a84d70b5445aae2d91573b
    37 schema:sameAs https://app.dimensions.ai/details/publication/pub.1107859896
    38 https://doi.org/10.1007/s00440-018-0881-6
    39 schema:sdDatePublished 2019-04-10T13:27
    40 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    41 schema:sdPublisher Naadf70b1b5d642d493b726623c3b8114
    42 schema:url https://link.springer.com/10.1007%2Fs00440-018-0881-6
    43 sgo:license sg:explorer/license/
    44 sgo:sdDataset articles
    45 rdf:type schema:ScholarlyArticle
    46 N24e3425afcaf4038a61329c6a75df637 schema:name doi
    47 schema:value 10.1007/s00440-018-0881-6
    48 rdf:type schema:PropertyValue
    49 N333b2216adb94dccbd53c0d8c4fb6338 schema:name dimensions_id
    50 schema:value pub.1107859896
    51 rdf:type schema:PropertyValue
    52 Naadf70b1b5d642d493b726623c3b8114 schema:name Springer Nature - SN SciGraph project
    53 rdf:type schema:Organization
    54 Nba9dbf91b0154068ae785755fa498e30 rdf:first sg:person.011242605267.32
    55 rdf:rest Nc1b4849729ec422e8cccdb58cad05e9d
    56 Nc08ed664bfe040efb67c78c60e0827c6 schema:affiliation https://www.grid.ac/institutes/grid.263306.2
    57 schema:familyName Holroyd
    58 schema:givenName Alexander E.
    59 rdf:type schema:Person
    60 Nc115fb0ca0a84d70b5445aae2d91573b schema:name readcube_id
    61 schema:value 0423568f07697f54c73cac9d7bac1cc7a92b05ce0c4177bf613d817eef727893
    62 rdf:type schema:PropertyValue
    63 Nc1b4849729ec422e8cccdb58cad05e9d rdf:first sg:person.0745225500.62
    64 rdf:rest rdf:nil
    65 Nc4f3d38977794620ac683943a36eaabb rdf:first Nc08ed664bfe040efb67c78c60e0827c6
    66 rdf:rest Nba9dbf91b0154068ae785755fa498e30
    67 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    68 schema:name Mathematical Sciences
    69 rdf:type schema:DefinedTerm
    70 anzsrc-for:0104 schema:inDefinedTermSet anzsrc-for:
    71 schema:name Statistics
    72 rdf:type schema:DefinedTerm
    73 sg:grant.2776052 http://pending.schema.org/fundedItem sg:pub.10.1007/s00440-018-0881-6
    74 rdf:type schema:MonetaryGrant
    75 sg:journal.1053886 schema:issn 0178-8051
    76 1432-2064
    77 schema:name Probability Theory and Related Fields
    78 rdf:type schema:Periodical
    79 sg:person.011242605267.32 schema:affiliation https://www.grid.ac/institutes/grid.462063.5
    80 schema:familyName Marcovici
    81 schema:givenName Irène
    82 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011242605267.32
    83 rdf:type schema:Person
    84 sg:person.0745225500.62 schema:affiliation https://www.grid.ac/institutes/grid.4991.5
    85 schema:familyName Martin
    86 schema:givenName James B.
    87 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0745225500.62
    88 rdf:type schema:Person
    89 sg:pub.10.1007/978-3-540-89646-3_115 schema:sameAs https://app.dimensions.ai/details/publication/pub.1044187555
    90 https://doi.org/10.1007/978-3-540-89646-3_115
    91 rdf:type schema:CreativeWork
    92 sg:pub.10.1007/bf00539839 schema:sameAs https://app.dimensions.ai/details/publication/pub.1003004744
    93 https://doi.org/10.1007/bf00539839
    94 rdf:type schema:CreativeWork
    95 sg:pub.10.1007/bf00679137 schema:sameAs https://app.dimensions.ai/details/publication/pub.1011876319
    96 https://doi.org/10.1007/bf00679137
    97 rdf:type schema:CreativeWork
    98 sg:pub.10.1007/bf01614095 schema:sameAs https://app.dimensions.ai/details/publication/pub.1040866374
    99 https://doi.org/10.1007/bf01614095
    100 rdf:type schema:CreativeWork
    101 sg:pub.10.1007/bf01646324 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041031628
    102 https://doi.org/10.1007/bf01646324
    103 rdf:type schema:CreativeWork
    104 sg:pub.10.1007/bfb0070087 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033207441
    105 https://doi.org/10.1007/bfb0070087
    106 rdf:type schema:CreativeWork
    107 sg:pub.10.1023/a:1004823720305 schema:sameAs https://app.dimensions.ai/details/publication/pub.1031444084
    108 https://doi.org/10.1023/a:1004823720305
    109 rdf:type schema:CreativeWork
    110 sg:pub.10.1023/a:1004824203467 schema:sameAs https://app.dimensions.ai/details/publication/pub.1049184356
    111 https://doi.org/10.1023/a:1004824203467
    112 rdf:type schema:CreativeWork
    113 https://doi.org/10.1002/rsa.20281 schema:sameAs https://app.dimensions.ai/details/publication/pub.1004532402
    114 rdf:type schema:CreativeWork
    115 https://doi.org/10.1016/0001-8708(79)90040-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1045595030
    116 rdf:type schema:CreativeWork
    117 https://doi.org/10.1016/j.tcs.2014.09.009 schema:sameAs https://app.dimensions.ai/details/publication/pub.1036268022
    118 rdf:type schema:CreativeWork
    119 https://doi.org/10.1016/s0012-365x(97)00109-x schema:sameAs https://app.dimensions.ai/details/publication/pub.1023290641
    120 rdf:type schema:CreativeWork
    121 https://doi.org/10.1017/s0963548303006035 schema:sameAs https://app.dimensions.ai/details/publication/pub.1050562815
    122 rdf:type schema:CreativeWork
    123 https://doi.org/10.1103/physrevlett.51.853 schema:sameAs https://app.dimensions.ai/details/publication/pub.1060789596
    124 rdf:type schema:CreativeWork
    125 https://doi.org/10.1137/1110026 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062865876
    126 rdf:type schema:CreativeWork
    127 https://doi.org/10.1142/s0129054117500174 schema:sameAs https://app.dimensions.ai/details/publication/pub.1085554926
    128 rdf:type schema:CreativeWork
    129 https://doi.org/10.1214/12-aihp530 schema:sameAs https://app.dimensions.ai/details/publication/pub.1064392840
    130 rdf:type schema:CreativeWork
    131 https://doi.org/10.1214/14-aop918 schema:sameAs https://app.dimensions.ai/details/publication/pub.1064394516
    132 rdf:type schema:CreativeWork
    133 https://doi.org/10.1214/16-aap1190 schema:sameAs https://app.dimensions.ai/details/publication/pub.1064395532
    134 rdf:type schema:CreativeWork
    135 https://doi.org/10.1214/aoap/1177004698 schema:sameAs https://app.dimensions.ai/details/publication/pub.1064397888
    136 rdf:type schema:CreativeWork
    137 https://doi.org/10.1239/aap/1386857853 schema:sameAs https://app.dimensions.ai/details/publication/pub.1064441128
    138 rdf:type schema:CreativeWork
    139 https://www.grid.ac/institutes/grid.263306.2 schema:alternateName Seattle University
    140 schema:name Seattle, USA
    141 rdf:type schema:Organization
    142 https://www.grid.ac/institutes/grid.462063.5 schema:alternateName Institut Élie Cartan de Lorraine
    143 schema:name Institut Elie Cartan de Lorraine, Université de Lorraine, Campus Scientifique, BP 239, 54506, Vandoeuvre-lès-Nancy Cedex, France
    144 rdf:type schema:Organization
    145 https://www.grid.ac/institutes/grid.4991.5 schema:alternateName University of Oxford
    146 schema:name Department of Statistics, University of Oxford, 1 South Parks Road, OX1 3TG, Oxford, UK
    147 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...