Random Structures and Algorithms最新文献

{"title":"Dixon's asymptotic without the classification of finite simple groups","authors":"Sean Eberhard","doi":"10.1002/rsa.21205","DOIUrl":"https://doi.org/10.1002/rsa.21205","url":null,"abstract":"Without using the classification of finite simple groups (CFSG), we show that the probability that two random elements of <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21205-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S Subscript n\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:rsa:media:rsa21205:rsa21205-math-0001\" display=\"inline\" location=\"graphic/rsa21205-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S Subscript n\" data-semantic-type=\"subscript\"><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi></mrow><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></msub></mrow>$$ {S}_n $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> generate a primitive group smaller than <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/rsa21205-math-0002.png\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper A Subscript n\" data-semantic-type=\"subscript\"><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mrow size=\"s\"><mjx-mi data-semantic-annotation=\"c","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Greedy maximal independent sets via local limits","authors":"Michael Krivelevich, Tamás Mészáros, Peleg Michaeli, Clara Shikhelman","doi":"10.1002/rsa.21200","DOIUrl":"https://doi.org/10.1002/rsa.21200","url":null,"abstract":"The random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent to any previously added vertex. In this paper, we present a general framework for computing the asymptotic density of the random greedy independent set for sequences of (possibly random) graphs by employing a notion of local convergence. We use this framework to give straightforward proofs for results on previously studied families of graphs, like paths and binomial random graphs, and to study new ones, like random trees and sparse random planar graphs. We conclude by analysing the random greedy algorithm more closely when the base graph is a tree.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"248 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random graphs embeddable in order-dependent surfaces","authors":"Colin McDiarmid, Sophia Saller","doi":"10.1002/rsa.21199","DOIUrl":"https://doi.org/10.1002/rsa.21199","url":null,"abstract":"Given a ‘genus function’ <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0001\" display=\"inline\" location=\"graphic/rsa21199-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>g</mi>\u0000<mo>=</mo>\u0000<mi>g</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ g=g(n) $$</annotation>\u0000</semantics></math>, we let <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0002\" display=\"inline\" location=\"graphic/rsa21199-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mi mathvariant=\"script\">E</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>g</mi>\u0000</mrow>\u0000</msup>\u0000</mrow>\u0000$$ {mathcal{E}}^g $$</annotation>\u0000</semantics></math> be the class of all graphs <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0003\" display=\"inline\" location=\"graphic/rsa21199-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math> such that if <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0004\" display=\"inline\" location=\"graphic/rsa21199-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math> has order <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0005\" display=\"inline\" location=\"graphic/rsa21199-math-0005.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000$$ n $$</annotation>\u0000</semantics></math> (i.e., has <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0006\" display=\"inline\" location=\"graphic/rsa21199-math-0006.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000$$ n $$</annotation>\u0000</semantics></math> vertices) then it is embeddable in a surface of Euler genus at most <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0007\" display=\"inline\" location=\"graphic/rsa21199-math-0007.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>g</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ g(n) $$</annotation>\u0000</semantics></math>. Let the random graph <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0008\" display=\"inline\" location=\"graphic/rsa21199-math-0008.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>R</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {R}_n $$</annotation>\u0000</semantics></math> be sampled uniformly from the graphs in <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0009\" display=\"inline\" location=\"graphic/rsa21199-math-0009.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mi mathvariant=\"script\">E</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>g</mi>\u0000</mrow>\u0000</msup>\u0000</mrow>\u0000$$ {mathcal{E}}^g $$</annotation>\u0000</semantics></math> on vertex set <math altimg=\"urn:x-wiley:rsa:media:rsa21199:rsa21199-math-0010\" display=\"inline\" location=\"graphic/rsa21199-math-0010.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mo stretchy=\"false\">[</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">]</mo>\u0000<mo>=</mo>\u0000<mo stretchy=\"false\">{</mo>\u0000<mn>1</mn>\u0000<mo>,</mo>\u0000<mi>…</mi>\u0000<mo>,</mo>\u0000<mi>n</mi>\u0000<mo stretchy=\"false\">}</mo>\u0000</mrow>\u0000$","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"1 11-12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Small cycle structure for words in conjugation invariant random permutations","authors":"Mohamed Slim Kammoun, Mylène Maïda","doi":"10.1002/rsa.21203","DOIUrl":"https://doi.org/10.1002/rsa.21203","url":null,"abstract":"We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive cyclic simplifications, the word <math altimg=\"urn:x-wiley:rsa:media:rsa21203:rsa21203-math-0001\" display=\"inline\" location=\"graphic/rsa21203-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>w</mi>\u0000</mrow>\u0000$$ w $$</annotation>\u0000</semantics></math> still contains at least two different letters, then we get a universal limiting joint law for short cycles for the word in these permutations. These results can be seen as an extension of our previous work (Kammoun and Maïda. <i>Electron. Commun. Probab.</i> 2020;25:1-14.) from the product of permutations to any non-trivial word in the permutations and also as an extension of the results of Nica (<i>Random Struct. Algorithms</i>1994;5:703-730.) from uniform permutations to general conjugation invariant random permutations. In particular, we get optimal assumptions in the case of the commutator of two such random permutations.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"1 5-6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On random irregular subgraphs","authors":"Jacob Fox, Sammy Luo, Huy Tuan Pham","doi":"10.1002/rsa.21204","DOIUrl":"https://doi.org/10.1002/rsa.21204","url":null,"abstract":"Let <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0001\" display=\"inline\" location=\"graphic/rsa21204-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math> be a <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0002\" display=\"inline\" location=\"graphic/rsa21204-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>d</mi>\u0000</mrow>\u0000$$ d $$</annotation>\u0000</semantics></math>-regular graph on <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0003\" display=\"inline\" location=\"graphic/rsa21204-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000$$ n $$</annotation>\u0000</semantics></math> vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0004\" display=\"inline\" location=\"graphic/rsa21204-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>H</mi>\u0000<mo>=</mo>\u0000<mi>H</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>G</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ H=H(G) $$</annotation>\u0000</semantics></math>. Assign independently to each vertex <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0005\" display=\"inline\" location=\"graphic/rsa21204-math-0005.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>v</mi>\u0000</mrow>\u0000$$ v $$</annotation>\u0000</semantics></math> of <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0006\" display=\"inline\" location=\"graphic/rsa21204-math-0006.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math> a uniform random number <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0007\" display=\"inline\" location=\"graphic/rsa21204-math-0007.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>x</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>v</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo>∈</mo>\u0000<mo stretchy=\"false\">[</mo>\u0000<mn>0</mn>\u0000<mo>,</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">]</mo>\u0000</mrow>\u0000$$ x(v)in left[0,1right] $$</annotation>\u0000</semantics></math>, and an edge <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0008\" display=\"inline\" location=\"graphic/rsa21204-math-0008.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>u</mi>\u0000<mo>,</mo>\u0000<mi>v</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000</mrow>\u0000$$ left(u,vright) $$</annotation>\u0000</semantics></math> of <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0009\" display=\"inline\" location=\"graphic/rsa21204-math-0009.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math> is an edge of <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0010\" display=\"inline\" location=\"graphic/rsa21204-math-0010.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>H</mi>\u0000</mrow>\u0000$$ H $$</annotation>\u0000</semantics></math> if and only if <math altimg=\"urn:x-wiley:rsa:media:rsa21204:rsa21204-math-0011\" display=\"inline\" location=\"graphic/rsa21204-math-0011.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"276 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rainbow Hamilton cycles in random geometric graphs","authors":"Alan Frieze, Xavier Pérez-Giménez","doi":"10.1002/rsa.21201","DOIUrl":"https://doi.org/10.1002/rsa.21201","url":null,"abstract":"Let <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0001\" display=\"inline\" location=\"graphic/rsa21201-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000<mo>,</mo>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000</msub>\u0000<mo>,</mo>\u0000<mi>…</mi>\u0000<mo>,</mo>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {X}_1,{X}_2,dots, {X}_n $$</annotation>\u0000</semantics></math> be chosen independently and uniformly at random from the unit <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0002\" display=\"inline\" location=\"graphic/rsa21201-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>d</mi>\u0000</mrow>\u0000$$ d $$</annotation>\u0000</semantics></math>-dimensional cube <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0003\" display=\"inline\" location=\"graphic/rsa21201-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mo stretchy=\"false\">[</mo>\u0000<mn>0</mn>\u0000<mo>,</mo>\u0000<mn>1</mn>\u0000<mo stretchy=\"false\">]</mo>\u0000</mrow>\u0000<mrow>\u0000<mi>d</mi>\u0000</mrow>\u0000</msup>\u0000</mrow>\u0000$$ {left[0,1right]}^d $$</annotation>\u0000</semantics></math>. Let <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0004\" display=\"inline\" location=\"graphic/rsa21201-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>r</mi>\u0000</mrow>\u0000$$ r $$</annotation>\u0000</semantics></math> be given and let <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0005\" display=\"inline\" location=\"graphic/rsa21201-math-0005.png\" overflow=\"scroll\">\u0000<mrow>\u0000<mi>𝒳</mi>\u0000<mo>=</mo>\u0000<mfenced close=\"}\" open=\"{\" separators=\"\">\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000<mo>,</mo>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000</msub>\u0000<mo>,</mo>\u0000<mi>…</mi>\u0000<mo>,</mo>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>n</mi>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000</mfenced>\u0000</mrow></math>. The random geometric graph <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0006\" display=\"inline\" location=\"graphic/rsa21201-math-0006.png\" overflow=\"scroll\">\u0000<mrow>\u0000<mi>G</mi>\u0000<mo>=</mo>\u0000<msub>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>𝒳</mi>\u0000<mo>,</mo>\u0000<mi>r</mi>\u0000</mrow>\u0000</msub>\u0000</mrow></math> has vertex set <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0007\" display=\"inline\" location=\"graphic/rsa21201-math-0007.png\" overflow=\"scroll\">\u0000<mrow>\u0000<mi>𝒳</mi>\u0000</mrow></math> and an edge <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0008\" display=\"inline\" location=\"graphic/rsa21201-math-0008.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>i</mi>\u0000</mrow>\u0000</msub>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>j</mi>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {X}_i{X}_j $$</annotation>\u0000</semantics></math> whenever <math altimg=\"urn:x-wiley:rsa:media:rsa21201:rsa21201-math-0009\" display=\"inline\" location=\"graphic/rsa21201-math-0009.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mo>‖</mo>\u0000<msub>\u0000<mrow>\u0000<mi>X</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>i</mi>\u0000</","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"61 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Frozen 1-RSB structure of the symmetric Ising perceptron","authors":"Will Perkins, Changji Xu","doi":"10.1002/rsa.21202","DOIUrl":"https://doi.org/10.1002/rsa.21202","url":null,"abstract":"We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1-RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"266 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wireless random-access networks with bipartite interference graphs","authors":"Sem C. Borst, Frank den Hollander, Francesca R. Nardi, Matteo Sfragara","doi":"10.1002/rsa.21198","DOIUrl":"https://doi.org/10.1002/rsa.21198","url":null,"abstract":"We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"2 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Note on down-set thresholds","authors":"Lutz Warnke","doi":"10.1002/rsa.21194","DOIUrl":"https://doi.org/10.1002/rsa.21194","url":null,"abstract":"Gunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up‐sets do not apply to down‐sets. In particular, for the down‐set of triangle‐free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down‐sets of ‐free graphs.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"2 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random perfect matchings in regular graphs","authors":"Bertille Granet, Felix Joos","doi":"10.1002/rsa.21172","DOIUrl":"https://doi.org/10.1002/rsa.21172","url":null,"abstract":"We prove that in all regular robust expanders <math altimg=\"urn:x-wiley:rsa:media:rsa21172:rsa21172-math-0001\" display=\"inline\" location=\"graphic/rsa21172-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math>, every edge is asymptotically equally likely contained in a uniformly chosen perfect matching <math altimg=\"urn:x-wiley:rsa:media:rsa21172:rsa21172-math-0002\" display=\"inline\" location=\"graphic/rsa21172-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>M</mi>\u0000</mrow>\u0000$$ M $$</annotation>\u0000</semantics></math>. We also show that given any fixed matching or spanning regular graph <math altimg=\"urn:x-wiley:rsa:media:rsa21172:rsa21172-math-0003\" display=\"inline\" location=\"graphic/rsa21172-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>N</mi>\u0000</mrow>\u0000$$ N $$</annotation>\u0000</semantics></math> in <math altimg=\"urn:x-wiley:rsa:media:rsa21172:rsa21172-math-0004\" display=\"inline\" location=\"graphic/rsa21172-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>G</mi>\u0000</mrow>\u0000$$ G $$</annotation>\u0000</semantics></math>, the random variable <math altimg=\"urn:x-wiley:rsa:media:rsa21172:rsa21172-math-0005\" display=\"inline\" location=\"graphic/rsa21172-math-0005.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mo stretchy=\"false\">|</mo>\u0000<mi>M</mi>\u0000<mo>∩</mo>\u0000<mi>E</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mi>N</mi>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo stretchy=\"false\">|</mo>\u0000</mrow>\u0000$$ mid Mcap E(N)mid $$</annotation>\u0000</semantics></math> is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"179 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}