We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step towards justification of the Bogomolny-Schmit heuristics.
{"title":"Fluctuations in the number of nodal domains","authors":"F. Nazarov, M. Sodin","doi":"10.1063/5.0018588","DOIUrl":"https://doi.org/10.1063/5.0018588","url":null,"abstract":"We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. \u0000Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step towards justification of the Bogomolny-Schmit heuristics.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86353861","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}
Given a sequence of Boolean functions ( (f_n)_{n geq 1} ), ( f_n colon { 0,1 }^{n} to { 0,1 }), and a sequence ( (X^{(n)})_{ngeq 1} ) of continuous time ( p_n )-biased random walks ( X^{(n)} = (X_t^{(n)})_{t geq 0}) on ( { 0,1 }^{n} ), let ( C_n ) be the (random) number of times in ( (0,1) ) at which the process ( (f_n(X_t))_{t geq 0} ) changes its value. In cite{js2006}, the authors conjectured that if ( (f_n)_{n geq 1} ) is non-degenerate, transitive and satisfies ( lim_{n to infty} mathbb{E}[C_n] = infty), then ( (C_n)_{n geq 1} ) is tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.
{"title":"A tame sequence of transitive Boolean functions","authors":"M. P. Forsström","doi":"10.1214/20-ecp366","DOIUrl":"https://doi.org/10.1214/20-ecp366","url":null,"abstract":"Given a sequence of Boolean functions ( (f_n)_{n geq 1} ), ( f_n colon { 0,1 }^{n} to { 0,1 }), and a sequence ( (X^{(n)})_{ngeq 1} ) of continuous time ( p_n )-biased random walks ( X^{(n)} = (X_t^{(n)})_{t geq 0}) on ( { 0,1 }^{n} ), let ( C_n ) be the (random) number of times in ( (0,1) ) at which the process ( (f_n(X_t))_{t geq 0} ) changes its value. In cite{js2006}, the authors conjectured that if ( (f_n)_{n geq 1} ) is non-degenerate, transitive and satisfies ( lim_{n to infty} mathbb{E}[C_n] = infty), then ( (C_n)_{n geq 1} ) is tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87333413","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}
We study the spectral norm of matrix random lifts $A^{(k,pi)}$ for a given $ntimes n$ matrix $A$ and $kge 2$, which is a random symmetric $kntimes kn$ matrix whose $ktimes k$ blocks are obtained by multiplying $A_{ij}$ by a $ktimes k$ matrix drawn independently from a distribution $pi$ supported on $ktimes k$ matrices with spectral norm at most $1$. Assuming that $mathbb{E}_pi X = 0$, we prove that [mathbb{E} |A^{(k,pi)}|lesssim max_{i}sqrt{sum_j A_{ij}^2}+max_{ij}|A_{ij}|sqrt{log (kn)}.] This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative $sqrt{log n}$ factor in the Non-Commutative Khintchine inequality can be removed. We also show an application on random $k$-lifts of graphs (each vertex of the graph is replaced with $k$ vertices, and each edge is replaced with a random bipartite matching between the two sets of $k$ vertices each). We prove an upper bound of $2(1+epsilon)sqrt{Delta}+O(sqrt{log(kn)})$ on the new eigenvalues for random $k$-lifts of a fixed $G = (V,E)$ with $|V| = n$ and maximum degree $Delta$, compared to the previous result of $O(sqrt{Deltalog(kn)})$ by Oliveira [Oli09] and the recent breakthrough by Bordenave and Collins [BC19] which gives $2sqrt{Delta-1} + o(1)$ as $krightarrowinfty$ for $Delta$-regular graph $G$.
{"title":"The spectral norm of random lifts of matrices","authors":"A. Bandeira, Yunzi Ding","doi":"10.1214/21-ecp415","DOIUrl":"https://doi.org/10.1214/21-ecp415","url":null,"abstract":"We study the spectral norm of matrix random lifts $A^{(k,pi)}$ for a given $ntimes n$ matrix $A$ and $kge 2$, which is a random symmetric $kntimes kn$ matrix whose $ktimes k$ blocks are obtained by multiplying $A_{ij}$ by a $ktimes k$ matrix drawn independently from a distribution $pi$ supported on $ktimes k$ matrices with spectral norm at most $1$. Assuming that $mathbb{E}_pi X = 0$, we prove that [mathbb{E} |A^{(k,pi)}|lesssim max_{i}sqrt{sum_j A_{ij}^2}+max_{ij}|A_{ij}|sqrt{log (kn)}.] This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative $sqrt{log n}$ factor in the Non-Commutative Khintchine inequality can be removed. We also show an application on random $k$-lifts of graphs (each vertex of the graph is replaced with $k$ vertices, and each edge is replaced with a random bipartite matching between the two sets of $k$ vertices each). We prove an upper bound of $2(1+epsilon)sqrt{Delta}+O(sqrt{log(kn)})$ on the new eigenvalues for random $k$-lifts of a fixed $G = (V,E)$ with $|V| = n$ and maximum degree $Delta$, compared to the previous result of $O(sqrt{Deltalog(kn)})$ by Oliveira [Oli09] and the recent breakthrough by Bordenave and Collins [BC19] which gives $2sqrt{Delta-1} + o(1)$ as $krightarrowinfty$ for $Delta$-regular graph $G$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74705329","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}
We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020+) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.
{"title":"Moderate Deviation estimates for Nodal Lengths\u0000of Random Spherical Harmonics","authors":"C. Macci, Maurizia Rossi, Anna Todino","doi":"10.30757/alea.v18-11","DOIUrl":"https://doi.org/10.30757/alea.v18-11","url":null,"abstract":"We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020+) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88020776","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}
We study systems of three interacting particles, in which drifts and variances are assigned by rank. These systems are "degenerate": the variances corresponding to one or two ranks can vanish, so the corresponding ranked motions become ballistic rather than diffusive. Depending on which ranks are allowed to "go ballistic", the systems exhibit markedly different behavior which we study in some detail. Also studied are stability properties for the resulting planar process of gaps between successive ranks.
{"title":"Degenerate competing three-particle systems","authors":"Tomoyuki Ichiba, I. Karatzas","doi":"10.3150/21-bej1411","DOIUrl":"https://doi.org/10.3150/21-bej1411","url":null,"abstract":"We study systems of three interacting particles, in which drifts and\u0000variances are assigned by rank. These systems are \"degenerate\": the variances\u0000corresponding to one or two ranks can vanish, so the corresponding ranked\u0000motions become ballistic rather than diffusive. Depending on which ranks are\u0000allowed to \"go ballistic\", the systems exhibit markedly different behavior\u0000which we study in some detail. Also studied are stability properties for the\u0000resulting planar process of gaps between successive ranks.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87478385","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}
Valentin Garino, I. Nourdin, D. Nualart, Majid Salamat
Consider a moving average process $X$ of the form $X(t)=int_{-infty}^t x(t-u)dZ_u$, $tgeq 0$, where $Z$ is a (non Gaussian) Hermite process of order $qgeq 2$ and $x:mathbb{R}_+tomathbb{R}$ is sufficiently integrable. This paper investigates the fluctuations, as $Ttoinfty$, of integral functionals of the form $tmapsto int_0^{Tt }P(X(s))ds$, in the case where $P$ is any given polynomial function. It extends a study initiated in Tran (2018), where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.
{"title":"Limit theorems for integral functionals of Hermite-driven processes","authors":"Valentin Garino, I. Nourdin, D. Nualart, Majid Salamat","doi":"10.3150/20-BEJ1291","DOIUrl":"https://doi.org/10.3150/20-BEJ1291","url":null,"abstract":"Consider a moving average process $X$ of the form $X(t)=int_{-infty}^t x(t-u)dZ_u$, $tgeq 0$, where $Z$ is a (non Gaussian) Hermite process of order $qgeq 2$ and $x:mathbb{R}_+tomathbb{R}$ is sufficiently integrable. This paper investigates the fluctuations, as $Ttoinfty$, of integral functionals of the form $tmapsto int_0^{Tt }P(X(s))ds$, in the case where $P$ is any given polynomial function. It extends a study initiated in Tran (2018), where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77453127","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}
In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable Levy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.
{"title":"Diffusions on a space of interval partitions: construction from Bertoin’s ${tt BES}_{0}(d)$, $din (0,1)$","authors":"Matthias Winkel","doi":"10.1214/20-ecp355","DOIUrl":"https://doi.org/10.1214/20-ecp355","url":null,"abstract":"In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable Levy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85920076","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}
Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.
{"title":"Approximations of the ruin probability in a discrete time risk model","authors":"David J. Santana, Luis Rincón","doi":"10.15559/20-vmsta158","DOIUrl":"https://doi.org/10.15559/20-vmsta158","url":null,"abstract":"Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72852881","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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