Schramm's Locality Conjecture asserts that the value of the critical percolation parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this note, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon.
{"title":"Locality of percolation for graphs with polynomial growth","authors":"D. Contreras, S'ebastien Martineau, V. Tassion","doi":"10.1214/22-ecp508","DOIUrl":"https://doi.org/10.1214/22-ecp508","url":null,"abstract":"Schramm's Locality Conjecture asserts that the value of the critical percolation parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this note, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43120874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a sharp upper bound for the number of high degree differences in bipartite graphs: let ( U, V, E ) be a bipartite graph with U = { u 1 , u 2 , . . . , u n } and V = { v 1 , v 2 , . . . , v n } ; for n ≥ k > n 2 we show that As a direct application we show a slightly stronger, probabilistic version of this theorem and thus confirm the Burdzy–Pitman conjecture about the maximal spread of coherent and independent distributions.
我们证明了二部图中高阶差数的一个尖锐上界:设(U, V, E)是一个U = {U 1, U 2，…的二部图。， u n}和V = {v1, v2，…， v n};作为一个直接应用，我们给出了这个定理的一个稍微强一点的概率版本，从而证实了关于相干和独立分布的最大扩展的Burdzy-Pitman猜想。
{"title":"A combinatorial proof of the Burdzy–Pitman conjecture","authors":"Stanisław Cichomski, F. Petrov","doi":"10.1214/23-ecp512","DOIUrl":"https://doi.org/10.1214/23-ecp512","url":null,"abstract":"We prove a sharp upper bound for the number of high degree differences in bipartite graphs: let ( U, V, E ) be a bipartite graph with U = { u 1 , u 2 , . . . , u n } and V = { v 1 , v 2 , . . . , v n } ; for n ≥ k > n 2 we show that As a direct application we show a slightly stronger, probabilistic version of this theorem and thus confirm the Burdzy–Pitman conjecture about the maximal spread of coherent and independent distributions.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41381317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.
{"title":"Introducing smooth amnesia to the memory of the Elephant Random Walk","authors":"Lucile Laulin","doi":"10.1214/22-ecp495","DOIUrl":"https://doi.org/10.1214/22-ecp495","url":null,"abstract":"This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42606328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble (cid:80) i (cid:54) = j f ( L N ( θ i − θ j )) for arbitrary β > 0 in the mesoscopic regime L N = O ( N 2 / 3 − (cid:15) ) . In addition, we obtain similar results for bipartite statistics.
{"title":"Central limit theorem for CβE pair dependent statistics in mesoscopic regime","authors":"A. Aguirre, A. Soshnikov","doi":"10.1214/22-ecp481","DOIUrl":"https://doi.org/10.1214/22-ecp481","url":null,"abstract":"We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble (cid:80) i (cid:54) = j f ( L N ( θ i − θ j )) for arbitrary β > 0 in the mesoscopic regime L N = O ( N 2 / 3 − (cid:15) ) . In addition, we obtain similar results for bipartite statistics.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43971099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum: Asymptotic results for empirical measures of weighted sums of independent random variables","authors":"B. Bercu, W. Bryc","doi":"10.1214/22-ecp464","DOIUrl":"https://doi.org/10.1214/22-ecp464","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49238290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simple halfspace depth","authors":"P. Laketa, D. Pokorný, Stanislav Nagy","doi":"10.1214/22-ecp503","DOIUrl":"https://doi.org/10.1214/22-ecp503","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48019149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.
{"title":"Distribution dependent SDEs for Navier-Stokes type equations","authors":"Fengchao Wang","doi":"10.1214/22-ecp479","DOIUrl":"https://doi.org/10.1214/22-ecp479","url":null,"abstract":"To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44821491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a 2-dimensional soft random geometric graph G ( λ, s, φ ), obtained by placing a Poisson( λs 2 ) number of vertices uniformly at random in a square of side s , with edges placed between each pair x, y of vertices with probability φ ( (cid:107) x − y (cid:107) ), where φ : R + → [0 , 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G ( λ, s, φ ) in the large- s limit with ( λ, φ ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λ c ( φ ).
{"title":"Giant component of the soft random geometric graph","authors":"M. Penrose","doi":"10.1214/22-ecp491","DOIUrl":"https://doi.org/10.1214/22-ecp491","url":null,"abstract":"Consider a 2-dimensional soft random geometric graph G ( λ, s, φ ), obtained by placing a Poisson( λs 2 ) number of vertices uniformly at random in a square of side s , with edges placed between each pair x, y of vertices with probability φ ( (cid:107) x − y (cid:107) ), where φ : R + → [0 , 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G ( λ, s, φ ) in the large- s limit with ( λ, φ ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λ c ( φ ).","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42670191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.
{"title":"McKean-Vlasov equations with positive feedback through elastic stopping times","authors":"B. Hambly, Julian Meier","doi":"10.1214/22-ecp482","DOIUrl":"https://doi.org/10.1214/22-ecp482","url":null,"abstract":"We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48157972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization n 1 / 6 ). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.
{"title":"A random graph of moderate density","authors":"Á. Backhausz, T. F. Móri","doi":"10.1214/21-ecp444","DOIUrl":"https://doi.org/10.1214/21-ecp444","url":null,"abstract":"We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization n 1 / 6 ). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46783625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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