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We assign the pair <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\left({\\overline{n } }^{*},{\\overline{v } }^{*}\\right)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mfenced>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mover>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo>\u00af<\/mml:mo>\n                        <\/mml:mover>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mrow\/>\n                        <mml:mo>\u2217<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mover>\n                          <mml:mi>v<\/mml:mi>\n                          <mml:mo>\u00af<\/mml:mo>\n                        <\/mml:mover>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mrow\/>\n                        <mml:mo>\u2217<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mfenced>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of <jats:italic>average corner degrees<\/jats:italic> (Domokos et al. in A two-vertex theorem for normal tilings. Aequat Math <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"https:\/\/doi.org\/10.1007\/s00010-022-00888-0\">https:\/\/doi.org\/10.1007\/s00010-022-00888-0<\/jats:ext-link>, 2022) to each crack pattern and we define two local, random evolutionary steps <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> and <jats:italic>R<\/jats:italic><jats:sub>1<\/jats:sub>, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\left({\\overline{n } }^{*},{\\overline{v } }^{*}\\right)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mfenced>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mover>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo>\u00af<\/mml:mo>\n                        <\/mml:mover>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mrow\/>\n                        <mml:mo>\u2217<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mover>\n                          <mml:mi>v<\/mml:mi>\n                          <mml:mo>\u00af<\/mml:mo>\n                        <\/mml:mover>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mrow\/>\n                        <mml:mo>\u2217<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mfenced>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> plane. We prove the existence of limit points for several types of trajectories. Also, we prove that <jats:italic>cell<\/jats:italic><jats:italic>density<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\\overline{\\rho }= \\frac{{\\overline{v } }^{*}}{{\\overline{n } }^{*}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mover>\n                      <mml:mi>\u03c1<\/mml:mi>\n                      <mml:mo>\u00af<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mfrac>\n                      <mml:msup>\n                        <mml:mrow>\n                          <mml:mover>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo>\u00af<\/mml:mo>\n                          <\/mml:mover>\n                        <\/mml:mrow>\n                        <mml:mrow>\n                          <mml:mrow\/>\n                          <mml:mo>\u2217<\/mml:mo>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                      <mml:msup>\n                        <mml:mrow>\n                          <mml:mover>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\u00af<\/mml:mo>\n                          <\/mml:mover>\n                        <\/mml:mrow>\n                        <mml:mrow>\n                          <mml:mrow\/>\n                          <mml:mo>\u2217<\/mml:mo>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                    <\/mml:mfrac>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>  increases monotonically under any admissible trajectory.<\/jats:p>","DOI":"10.1007\/s10100-022-00838-w","type":"journal-article","created":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T17:05:17Z","timestamp":1672333517000},"page":"83-94","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A discrete time evolution model for fracture networks"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8676-6829","authenticated-orcid":false,"given":"G\u00e1bor","family":"Domokos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Krisztina","family":"Reg\u0151s","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,29]]},"reference":[{"key":"838_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-1599-7","volume-title":"Fractures and fracture networks","author":"PM Adler","year":"1999","unstructured":"Adler PM, Thovert JF (1999) Fractures and fracture networks. 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