Pub Date : 2019-12-23DOI: 10.1017/S0963548320000024
Michael C. H. Choi, P. Patie
Abstract In this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.
{"title":"Analysis of non-reversible Markov chains via similarity orbits","authors":"Michael C. H. Choi, P. Patie","doi":"10.1017/S0963548320000024","DOIUrl":"https://doi.org/10.1017/S0963548320000024","url":null,"abstract":"Abstract In this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87054542","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}
Pub Date : 2019-11-11DOI: 10.1017/S0963548320000218
Ryan Alweiss, Chady Ben Hamida, Xiaoyu He, Alexander Moreira
Abstract Given a fixed graph H, a real number p (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2 n in G(n, 1/2) in nδ queries.
{"title":"On the subgraph query problem","authors":"Ryan Alweiss, Chady Ben Hamida, Xiaoyu He, Alexander Moreira","doi":"10.1017/S0963548320000218","DOIUrl":"https://doi.org/10.1017/S0963548320000218","url":null,"abstract":"Abstract Given a fixed graph H, a real number p (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2 n in G(n, 1/2) in nδ queries.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77920252","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}
Pub Date : 2019-10-28DOI: 10.1017/S0963548320000462
Huiqiu Lin, Bo Ning, Baoyindureng Wu
Abstract Bollobás and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${rm{lambda }}_1^2(G) + {rm{lambda }}_2^2(G) le (r - 1)/r cdot 2m$, where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdös and Nosal respectively, we prove that every non-bipartite graph of order and size contains a triangle if one of the following is true: (i) ${{rm{lambda }}_1}(G) ge sqrt {m - 1} $ and $G ne {C_5} cup (n - 5){K_1}$, and (ii) ${{rm{lambda }}_1}(G) ge {{rm{lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$, where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.
摘要Bollobás和Nikiforov (J. Combin。理论SerB. 97(2007) 859-865)推测如下。如果G是一个至少有r+1个顶点和m条边的无Kr+1的图，则${rm{lambda }}_1^2(G) + {rm{lambda }}_2^2(G) le (r - 1)/r cdot 2m$，其中λ1 (G)和λ2 (G)分别是邻接矩阵a (G)的最大和第二大特征值。本文利用双随机矩阵理论的工具，证实了r=2情况下的猜想，并刻画了极值图的所有族。在分别由Erdös和Nosal给出的经典定理的启发下，我们证明了如果下列条件之一成立，则每个阶数和大小的非二部图都包含一个三角形:(i) ${{rm{lambda }}_1}(G) ge sqrt {m - 1} $和$G ne {C_5} cup (n - 5){K_1}$，以及(ii) ${{rm{lambda }}_1}(G) ge {{rm{lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$和$G ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$，其中$S({K_{[(n - 1)/2],[(n - 1)/2]}})$是通过细分一条边从${K_{[(n - 1)/2],[(n - 1)/2]}}$得到的。这两种情况都是最好的。最后，我们提出了一些有待解决的问题。
{"title":"Eigenvalues and triangles in graphs","authors":"Huiqiu Lin, Bo Ning, Baoyindureng Wu","doi":"10.1017/S0963548320000462","DOIUrl":"https://doi.org/10.1017/S0963548320000462","url":null,"abstract":"Abstract Bollobás and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${rm{lambda }}_1^2(G) + {rm{lambda }}_2^2(G) le (r - 1)/r cdot 2m$, where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdös and Nosal respectively, we prove that every non-bipartite graph of order and size contains a triangle if one of the following is true: (i) ${{rm{lambda }}_1}(G) ge sqrt {m - 1} $ and $G ne {C_5} cup (n - 5){K_1}$, and (ii) ${{rm{lambda }}_1}(G) ge {{rm{lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$, where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82677562","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}
Pub Date : 2019-10-08DOI: 10.1017/S0963548319000269
Bo Ning, Xing Peng
Abstract The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.
著名的Turán条路径数Erdős-Gallai定理指出，每一个有n个顶点和m条边的图都包含一条至少有(2m)/n条边的路径。在这篇笔记中，我们首先通过证明每个图G包含至少有$${{(s + 1){N_{s + 1}}(G)} over {{N_s}(G)}} + s - 1$$条边的路径来建立Erdős-Gallai定理的一个简单而新颖的扩展，其中Nj(G)表示当1≤j≤ω(G)时，G中j-团的个数。我们还构造了一组图，证明我们的推广改进了Erdős-Gallai定理给出的估计。例如，在应用中，我们证明了[20]的主要结果可以很容易地从这个扩展中推导出来，这些结果是关于在一个n顶点图中不存在具有r顶点的路径(并且不存在长度至少为c的循环)的s-团的最大可能数量。实际上，为了证明这些结果，Luo[20]推广了Kopylov的经典定理，并建立了周长小于c的n顶点2连通图中s-团数的紧上界。对于周长小于c且最小度较大的n顶点2连通图，我们证明了类似的结果。最后，我们将所得结果应用于图中关于连续循环长度的谱极值图论问题。
{"title":"Extensions of the Erdős–Gallai theorem and Luo’s theorem","authors":"Bo Ning, Xing Peng","doi":"10.1017/S0963548319000269","DOIUrl":"https://doi.org/10.1017/S0963548319000269","url":null,"abstract":"Abstract The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91385901","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}
Pub Date : 2019-10-07DOI: 10.1017/s096354832300007x
B. Schülke
We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$ -uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle. In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$ -uniform hypergraphs by proving a $3$ -uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.
{"title":"A pair degree condition for Hamiltonian cycles in 3-uniform hypergraphs","authors":"B. Schülke","doi":"10.1017/s096354832300007x","DOIUrl":"https://doi.org/10.1017/s096354832300007x","url":null,"abstract":"\u0000 We prove a new sufficient pair degree condition for tight Hamiltonian cycles in \u0000 \u0000 \u0000 \u0000$3$\u0000\u0000 \u0000 -uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.\u0000 In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in \u0000 \u0000 \u0000 \u0000$3$\u0000\u0000 \u0000 -uniform hypergraphs by proving a \u0000 \u0000 \u0000 \u0000$3$\u0000\u0000 \u0000 -uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74564236","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}
Pub Date : 2019-10-03DOI: 10.1017/s0963548323000159
Ádám Timár
We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.
{"title":"Unimodular random one-ended planar graphs are sofic","authors":"Ádám Timár","doi":"10.1017/s0963548323000159","DOIUrl":"https://doi.org/10.1017/s0963548323000159","url":null,"abstract":"\u0000 We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75298412","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}
Pub Date : 2019-09-30DOI: 10.1017/S0963548319000233
Shachar Sapir, A. Shapira
Abstract The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.
{"title":"The Induced Removal Lemma in Sparse Graphs","authors":"Shachar Sapir, A. Shapira","doi":"10.1017/S0963548319000233","DOIUrl":"https://doi.org/10.1017/S0963548319000233","url":null,"abstract":"Abstract The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80088631","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}
Pub Date : 2019-09-12DOI: 10.1017/S0963548319000178
M. Alishahi, H. Hajiabolhassan
Abstract In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.
{"title":"On the Chromatic Number of Matching Kneser Graphs","authors":"M. Alishahi, H. Hajiabolhassan","doi":"10.1017/S0963548319000178","DOIUrl":"https://doi.org/10.1017/S0963548319000178","url":null,"abstract":"Abstract In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87476935","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}
Pub Date : 2019-08-09DOI: 10.1017/s0963548323000214
Agelos Georgakopoulos, C. Panagiotis
We introduce a formula for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice versa. We exploit this in both directions. We obtain the rigorous lower bound ${dot{p}_c}({mathbb{Z}}^3)gt 0.2522$ for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on $a({mathbb{Z}}^d)$ as $dto infty$ . Our formula remains valid if instead of lattice animals we enumerate certain subspecies called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $pin (0,1)$ .
{"title":"On the exponential growth rates of lattice animals and interfaces","authors":"Agelos Georgakopoulos, C. Panagiotis","doi":"10.1017/s0963548323000214","DOIUrl":"https://doi.org/10.1017/s0963548323000214","url":null,"abstract":"\u0000 We introduce a formula for translating any upper bound on the percolation threshold of a lattice \u0000 \u0000 \u0000 \u0000$G$\u0000\u0000 \u0000 into a lower bound on the exponential growth rate of lattice animals \u0000 \u0000 \u0000 \u0000$a(G)$\u0000\u0000 \u0000 and vice versa. We exploit this in both directions. We obtain the rigorous lower bound \u0000 \u0000 \u0000 \u0000${dot{p}_c}({mathbb{Z}}^3)gt 0.2522$\u0000\u0000 \u0000 for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on \u0000 \u0000 \u0000 \u0000$a({mathbb{Z}}^d)$\u0000\u0000 \u0000 as \u0000 \u0000 \u0000 \u0000$dto infty$\u0000\u0000 \u0000 . Our formula remains valid if instead of lattice animals we enumerate certain subspecies called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold.\u0000 Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of \u0000 \u0000 \u0000 \u0000$pin (0,1)$\u0000\u0000 \u0000 .","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76005074","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}
Pub Date : 2019-07-30DOI: 10.1017/s0963548323000093
Ander Lamaison
For a fixed infinite graph $H$ , we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-colouring of the edges of $K_{mathbb{N}}$ . This is called the Ramsey upper density of $H$ and was introduced by Erdős and Galvin in a restricted setting, and by DeBiasio and McKenney in general. Recently [4], the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimisation problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of $H$ , the number of components of $H$ and the expansion ratio $|N(I)|/|I|$ of the independent sets of $H$ .
{"title":"Ramsey upper density of infinite graphs","authors":"Ander Lamaison","doi":"10.1017/s0963548323000093","DOIUrl":"https://doi.org/10.1017/s0963548323000093","url":null,"abstract":"\u0000 For a fixed infinite graph \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 , we study the largest density of a monochromatic subgraph isomorphic to \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 that can be found in every two-colouring of the edges of \u0000 \u0000 \u0000 \u0000$K_{mathbb{N}}$\u0000\u0000 \u0000 . This is called the Ramsey upper density of \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 and was introduced by Erdős and Galvin in a restricted setting, and by DeBiasio and McKenney in general. Recently [4], the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimisation problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 , the number of components of \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 and the expansion ratio \u0000 \u0000 \u0000 \u0000$|N(I)|/|I|$\u0000\u0000 \u0000 of the independent sets of \u0000 \u0000 \u0000 \u0000$H$\u0000\u0000 \u0000 .","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80000257","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}
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