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Topology"],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$X_1,X_2, \\ldots $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be independent identically distributed random points in a convex polytopal domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$A \\subset \\mathbb {R}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>R<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Define the <jats:italic>largest nearest-neighbour link<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$L_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to be the smallest <jats:italic>r<\/jats:italic> such that every point of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathscr {X}_n:=\\{X_1,\\ldots ,X_n\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>{<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>X<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mo>\u2026<\/mml:mo>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>X<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>}<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has another such point within distance <jats:italic>r<\/jats:italic>. We obtain a strong law of large numbers for <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in the large-<jats:italic>n<\/jats:italic> limit. A related threshold, the <jats:italic>connectivity threshold<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$M_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, is the smallest <jats:italic>r<\/jats:italic> such that the random geometric graph <jats:inline-formula><jats:alternatives><jats:tex-math>$$G(\\mathscr {X}_n, r)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>r<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is connected (so <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_n \\le M_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>). We show that as <jats:inline-formula><jats:alternatives><jats:tex-math>$$n \\rightarrow \\infty $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>\u221e<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, almost surely <jats:inline-formula><jats:alternatives><jats:tex-math>$$nL_n^d\/\\log n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:msubsup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mo>log<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> tends to a limit that depends on the geometry of <jats:italic>A<\/jats:italic>, and <jats:inline-formula><jats:alternatives><jats:tex-math>$$nM_n^d\/\\log n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:msubsup>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mo>log<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> tends to the same limit. We derive these results via asymptotic lower bounds for <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and upper bounds for <jats:inline-formula><jats:alternatives><jats:tex-math>$$M_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that are applicable in a larger class of metric spaces satisfying certain regularity conditions.<\/jats:p>","DOI":"10.1007\/s41468-023-00154-5","type":"journal-article","created":{"date-parts":[[2023,12,16]],"date-time":"2023-12-16T19:01:36Z","timestamp":1702753296000},"page":"1723-1750","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Largest nearest-neighbour link and connectivity threshold in a polytopal random sample"],"prefix":"10.1007","volume":"8","author":[{"given":"Mathew D.","family":"Penrose","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2435-4615","authenticated-orcid":false,"given":"Xiaochuan","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7300-8412","authenticated-orcid":false,"given":"Frankie","family":"Higgs","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,12,16]]},"reference":[{"key":"154_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/1300000026","volume":"4","author":"F Baccelli","year":"2009","unstructured":"Baccelli, F., B\u0142aszczyszyn, B.: Stochastic geometry and wireless networks I: theory. 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