Abstract Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4‐cycle in the binomial random graph . We observe a new phenomenon in the theory of large deviations of subgraph counts. This phenomenon is that, in a certain (large) range of , the upper tail of the induced 4‐cycle does not admit a naive mean‐field approximation.
{"title":"The upper tail problem for induced 4‐cycles in sparse random graphs","authors":"Asaf Cohen Antonir","doi":"10.1002/rsa.21187","DOIUrl":"https://doi.org/10.1002/rsa.21187","url":null,"abstract":"Abstract Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4‐cycle in the binomial random graph . We observe a new phenomenon in the theory of large deviations of subgraph counts. This phenomenon is that, in a certain (large) range of , the upper tail of the induced 4‐cycle does not admit a naive mean‐field approximation.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136113970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we propose to study a general notion of a down‐up Markov chain for multifurcating trees with labeled leaves. We study in detail down‐up chains associated with the ‐model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018; Random Struct. Algoritm. 54 (2020), 745–769; Electron. J. Probab. 25 (2020), 1–46.) in the binary special cases. The technique we deploy utilizes the construction of a growth process and a down‐up Markov chain on trees with planar structure. Our construction ensures that natural projections of the down‐up chain are Markov chains in their own right. We establish label dynamics that at the same time preserve the labeled alpha‐gamma distribution and keep the branch points between the smallest labels for order time steps for all . We conjecture the existence of diffusive scaling limits generalizing the “Aldous diffusion” by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018.) as a continuum‐tree‐valued process and the “algebraic ‐Ford tree evolution” by Löhr et al. (Ann. Probab. 48 (2020), 2565–2590.) and by Nussbaumer and Winter (arXiv:2006.09316, 2020.) as a process in a space of algebraic trees.
{"title":"A down‐up chain with persistent labels on multifurcating trees","authors":"Frederik Sørensen","doi":"10.1002/rsa.21185","DOIUrl":"https://doi.org/10.1002/rsa.21185","url":null,"abstract":"Abstract In this article, we propose to study a general notion of a down‐up Markov chain for multifurcating trees with labeled leaves. We study in detail down‐up chains associated with the ‐model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018; Random Struct. Algoritm. 54 (2020), 745–769; Electron. J. Probab. 25 (2020), 1–46.) in the binary special cases. The technique we deploy utilizes the construction of a growth process and a down‐up Markov chain on trees with planar structure. Our construction ensures that natural projections of the down‐up chain are Markov chains in their own right. We establish label dynamics that at the same time preserve the labeled alpha‐gamma distribution and keep the branch points between the smallest labels for order time steps for all . We conjecture the existence of diffusive scaling limits generalizing the “Aldous diffusion” by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018.) as a continuum‐tree‐valued process and the “algebraic ‐Ford tree evolution” by Löhr et al. (Ann. Probab. 48 (2020), 2565–2590.) and by Nussbaumer and Winter (arXiv:2006.09316, 2020.) as a process in a space of algebraic trees.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry‐invariant? This question was answered for one‐ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.
{"title":"A full characterization of invariant embeddability of unimodular planar graphs","authors":"Ádám Timár, László Márton Tóth","doi":"10.1002/rsa.21188","DOIUrl":"https://doi.org/10.1002/rsa.21188","url":null,"abstract":"Abstract When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry‐invariant? This question was answered for one‐ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"100 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an oriented graph , let denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.
{"title":"Extremal results on feedback arc sets in digraphs","authors":"J. Fox, Z. Himwich, Nitya Mani","doi":"10.1002/rsa.21179","DOIUrl":"https://doi.org/10.1002/rsa.21179","url":null,"abstract":"For an oriented graph , let denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"41 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80416418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Dovgal, Élie de Panafieu, D. Ralaivaosaona, Vonjy Rasendrahasina, S. Wagner
It is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.
{"title":"The birth of the strong components","authors":"S. Dovgal, Élie de Panafieu, D. Ralaivaosaona, Vonjy Rasendrahasina, S. Wagner","doi":"10.1002/rsa.21176","DOIUrl":"https://doi.org/10.1002/rsa.21176","url":null,"abstract":"It is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"36 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89550802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the anisotropic random geometric graph model, vertices correspond to points drawn from a high‐dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős‐Rényi graph with the same edge probability. If is the number of vertices and is the vector of eigenvalues, Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017 shows that detection is possible when and impossible when . We show detection is impossible when , closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017.
{"title":"Threshold for detecting high dimensional geometry in anisotropic random geometric graphs","authors":"Matthew Brennan, Guy Bresler, Brice Huang","doi":"10.1002/rsa.21178","DOIUrl":"https://doi.org/10.1002/rsa.21178","url":null,"abstract":"Abstract In the anisotropic random geometric graph model, vertices correspond to points drawn from a high‐dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős‐Rényi graph with the same edge probability. If is the number of vertices and is the vector of eigenvalues, Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017 shows that detection is possible when and impossible when . We show detection is impossible when , closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"215 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136021126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Thomas Bläsius, Tobias Friedrich, Andreas Göbel, Jordi Levy, Ralf Rothenberger
Abstract Satisfiability is considered the canonical NP‐complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large‐scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real‐world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random ‐SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random ‐SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT‐solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time.
{"title":"The impact of heterogeneity and geometry on the proof complexity of random satisfiability","authors":"Thomas Bläsius, Tobias Friedrich, Andreas Göbel, Jordi Levy, Ralf Rothenberger","doi":"10.1002/rsa.21168","DOIUrl":"https://doi.org/10.1002/rsa.21168","url":null,"abstract":"Abstract Satisfiability is considered the canonical NP‐complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large‐scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real‐world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random ‐SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random ‐SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT‐solvers. In contrast, modeling locality with underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135155951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order , in contrast to the bound obtained from the Gaussian–Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function that arises in the representation of the disorder as for standard normal. Under an additional regularity assumption on , we further show that the variance is of order at most .
{"title":"Universality of superconcentration in the Sherrington–Kirkpatrick model","authors":"Wei-Kuo Chen, Wai-Kit Lam","doi":"10.1002/rsa.21183","DOIUrl":"https://doi.org/10.1002/rsa.21183","url":null,"abstract":"We study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order , in contrast to the bound obtained from the Gaussian–Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function that arises in the representation of the disorder as for standard normal. Under an additional regularity assumption on , we further show that the variance is of order at most .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"65 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78877180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, motivated by a problem of Scott [Surveys in combinatorics, 327 (2005), 95‐117.] and a conjecture of Lee et al. [Random Struct. Algorithm, 48 (2016), 147‐170.] we consider bisections of directed graphs. We prove that every directed graph with arcs and minimum semidegree at least admits a bisection in which at least arcs cross in each direction. This provides an optimal bound as well as a positive answer to a question of Hou and Wu [J. Comb. Theory B, 132 (2018), 107‐133.] in a stronger form.
在这篇文章中，受Scott [Surveys In combinatorics, 327(2005)， 95‐117]的一个问题的启发。和Lee等人的猜想[随机结构]。算法，48(2016)，147‐170。我们考虑有向图的等分。我们证明了每一个具有弧和最小半度的有向图至少存在一个在每个方向上至少有弧相交的平分。这提供了一个最优边界，以及一个积极的回答问题的侯和吴[J]。梳子。理论B, 32(2018)， 107‐133。以更强的形式。
{"title":"Optimal bisections of directed graphs","authors":"Guanwu Liu, Jie Ma, C. Zu","doi":"10.1002/rsa.21175","DOIUrl":"https://doi.org/10.1002/rsa.21175","url":null,"abstract":"In this article, motivated by a problem of Scott [Surveys in combinatorics, 327 (2005), 95‐117.] and a conjecture of Lee et al. [Random Struct. Algorithm, 48 (2016), 147‐170.] we consider bisections of directed graphs. We prove that every directed graph with arcs and minimum semidegree at least admits a bisection in which at least arcs cross in each direction. This provides an optimal bound as well as a positive answer to a question of Hou and Wu [J. Comb. Theory B, 132 (2018), 107‐133.] in a stronger form.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88861572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a general framework for approximations to decomposable combinatorial structures suitable to the situation where the size n$$ n $$ and the number of components k$$ k $$ are specified. In particular for assemblies, this involves a Poisson process, which, with the appropriate choice of parameter, may be viewed as an extension of saddlepoint approximation. We illustrate the use of the Poisson process description for assemblies by analyzing the component structure when the rank, defined as r:=n−k$$ r:= n-k $$ , is small relative to the size n$$ n $$ . There is near‐universal behavior, in the sense that, apart from degenerate cases where the exponential generating function has radius of convergence zero, we have for t∈(0,∞)$$ tin left(0,infty right) $$ and ℓ=1,2,…$$ ell =1,2,dots $$ : when r≍nα$$ rasymp {n}^{alpha } $$ for fixed α∈(ℓℓ+1,ℓ+1ℓ+2)$$ alpha in left(frac{ell }{ell +1},frac{ell +1}{ell +2}right) $$ , the size L1$$ {L}_1 $$ of the largest component converges in probability to ℓ+2$$ ell +2 $$ ; when r∼tnℓ/(ℓ+1)$$ rsim tkern0.3em {n}^{ell /left(ell +1right)} $$ , we have ℙ(L1∈{ℓ+1,ℓ+2})→1$$ mathbb{P}left({L}_1in left{ell +1,ell +2right}right)to 1 $$ , with the choice governed by a Poisson limit distribution for the number of components of size ℓ+2$$ ell +2 $$ . This was recently observed, for the case ℓ=1$$ ell =1 $$ and the special cases of permutations and set partitions, using Chen–Stein approximations for the indicators of attacks and alignments in rook placements.
{"title":"On the largest part size of low‐rank combinatorial assemblies","authors":"R. Arratia, S. Desalvo","doi":"10.1002/rsa.21132","DOIUrl":"https://doi.org/10.1002/rsa.21132","url":null,"abstract":"We give a general framework for approximations to decomposable combinatorial structures suitable to the situation where the size n$$ n $$ and the number of components k$$ k $$ are specified. In particular for assemblies, this involves a Poisson process, which, with the appropriate choice of parameter, may be viewed as an extension of saddlepoint approximation. We illustrate the use of the Poisson process description for assemblies by analyzing the component structure when the rank, defined as r:=n−k$$ r:= n-k $$ , is small relative to the size n$$ n $$ . There is near‐universal behavior, in the sense that, apart from degenerate cases where the exponential generating function has radius of convergence zero, we have for t∈(0,∞)$$ tin left(0,infty right) $$ and ℓ=1,2,…$$ ell =1,2,dots $$ : when r≍nα$$ rasymp {n}^{alpha } $$ for fixed α∈(ℓℓ+1,ℓ+1ℓ+2)$$ alpha in left(frac{ell }{ell +1},frac{ell +1}{ell +2}right) $$ , the size L1$$ {L}_1 $$ of the largest component converges in probability to ℓ+2$$ ell +2 $$ ; when r∼tnℓ/(ℓ+1)$$ rsim tkern0.3em {n}^{ell /left(ell +1right)} $$ , we have ℙ(L1∈{ℓ+1,ℓ+2})→1$$ mathbb{P}left({L}_1in left{ell +1,ell +2right}right)to 1 $$ , with the choice governed by a Poisson limit distribution for the number of components of size ℓ+2$$ ell +2 $$ . This was recently observed, for the case ℓ=1$$ ell =1 $$ and the special cases of permutations and set partitions, using Chen–Stein approximations for the indicators of attacks and alignments in rook placements.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"145 14 1","pages":"26 - 3"},"PeriodicalIF":1.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80219358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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