We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $Pin mathbb Z[x]$ that is a product of distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant $kappa_P>0$ such that [ frac{1}{sqrt{kappa_P N}}sum_{nleq N}f(P(n))xrightarrow{d}mathcal{N}(0,1), ] as $Nrightarrowinfty$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu. We also study large fluctuations of $sum_{nleq N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that [ Big|sum_{nleq N}f(n^2+1)Big|gg sqrt{N loglog N}. ] This matches the bound one expects from the law of iterated logarithm.
{"title":"Random Chowla's Conjecture for Rademacher Multiplicative Functions","authors":"Jake Chinis, Besfort Shala","doi":"arxiv-2409.05952","DOIUrl":"https://doi.org/arxiv-2409.05952","url":null,"abstract":"We study the distribution of partial sums of Rademacher random multiplicative\u0000functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a\u0000polynomial $Pin mathbb Z[x]$ that is a product of distinct linear factors or\u0000an irreducible quadratic satisfying a natural condition, there exists a\u0000constant $kappa_P>0$ such that [ frac{1}{sqrt{kappa_P N}}sum_{nleq\u0000N}f(P(n))xrightarrow{d}mathcal{N}(0,1), ] as $Nrightarrowinfty$, where convergence is in distribution to a standard\u0000(real) Gaussian. This confirms a conjecture of Najnudel and addresses a\u0000question of Klurman-Shkredov-Xu. We also study large fluctuations of $sum_{nleq N}f(n^2+1)$ and show that\u0000there almost surely exist arbitrarily large values of $N$ such that [\u0000Big|sum_{nleq N}f(n^2+1)Big|gg sqrt{N loglog N}. ] This matches the\u0000bound one expects from the law of iterated logarithm.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212048","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 consider asymmetric simple exclusion processes with $N$ particles on the one-dimensional discrete torus with $L$ sites with following properties: (i) nearest-neighbor jumps on the torus, (ii) the jump rates depend only on the distance to the next particle in the direction of the jump, (iii) the jump rates are independent of $N$ and $L$. For measures with a long-range two-body interaction potential that depends only on the distance between neighboring particles we prove a relation between the interaction potential and particle jump rates that is necessary and sufficient for the measure to be invariant for the process. The normalization of the measure and the stationary current are computed both for finite $L$ and $N$ and in the thermodynamic limit. For a finitely many particles that evolve on $mathbb{Z}$ with totally asymmetric jumps it is proved, using reverse duality, that a certain family of nonstationary measures with a microscopic shock and antishock evolves into a convex combination of such measures with weights given by random walk transition probabilities. On macroscopic scale this domain random walk is a travelling wave phenomenon tantamount to phase separation with a stable shock and stable antishock. Various potential applications of this result and open questions are outlined.
{"title":"Asymmetric exclusion process with long-range interactions","authors":"V. Belitsky, N. P. N. Ngoc, G. M. Schütz","doi":"arxiv-2409.05017","DOIUrl":"https://doi.org/arxiv-2409.05017","url":null,"abstract":"We consider asymmetric simple exclusion processes with $N$ particles on the\u0000one-dimensional discrete torus with $L$ sites with following properties: (i)\u0000nearest-neighbor jumps on the torus, (ii) the jump rates depend only on the\u0000distance to the next particle in the direction of the jump, (iii) the jump\u0000rates are independent of $N$ and $L$. For measures with a long-range two-body\u0000interaction potential that depends only on the distance between neighboring\u0000particles we prove a relation between the interaction potential and particle\u0000jump rates that is necessary and sufficient for the measure to be invariant for\u0000the process. The normalization of the measure and the stationary current are\u0000computed both for finite $L$ and $N$ and in the thermodynamic limit. For a\u0000finitely many particles that evolve on $mathbb{Z}$ with totally asymmetric\u0000jumps it is proved, using reverse duality, that a certain family of\u0000nonstationary measures with a microscopic shock and antishock evolves into a\u0000convex combination of such measures with weights given by random walk\u0000transition probabilities. On macroscopic scale this domain random walk is a\u0000travelling wave phenomenon tantamount to phase separation with a stable shock\u0000and stable antishock. Various potential applications of this result and open\u0000questions are outlined.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212152","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 show that for every ergodic and aperiodic probability preserving system $(X,mathcal{B},m,T)$, there exists $f:Xto mathbb{Z}^d$, whose corresponding cocycle satisfies the d-dimensional local central limit theorem. We use the 2-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence of polynomial multiple averages of non-commuting zero entropy transformations.
我们证明，对于每个遍历和非周期性概率保全系统$(X,mathcal{B},m,T)$，存在$f:Xto mathbb{Z}^d$，其相应的循环满足 d 维局部中心极限定理。我们利用二维结果来解决黄、邵和叶以及弗兰齐基纳基斯和霍斯特关于非交换零熵变换的多项式多重平均的不收敛问题。
{"title":"Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages","authors":"Zemer Kosloff, Shrey Sanadhya","doi":"arxiv-2409.05087","DOIUrl":"https://doi.org/arxiv-2409.05087","url":null,"abstract":"We show that for every ergodic and aperiodic probability preserving system\u0000$(X,mathcal{B},m,T)$, there exists $f:Xto mathbb{Z}^d$, whose corresponding\u0000cocycle satisfies the d-dimensional local central limit theorem. We use the 2-dimensional result to resolve a question of Huang, Shao and Ye\u0000and Franzikinakis and Host regarding non-convergence of polynomial multiple\u0000averages of non-commuting zero entropy transformations.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212161","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}
The birth-death process is a special type of continuous-time Markov chain with index set $mathbb{N}$. Its resolvent matrix can be fully characterized by a set of parameters $(gamma, beta, nu)$, where $gamma$ and $beta$ are non-negative constants, and $nu$ is a positive measure on $mathbb{N}$. By employing the Ray-Knight compactification, the birth-death process can be realized as a c`adl`ag process with strong Markov property on the one-point compactification space $overline{mathbb{N}}_{partial}$, which includes an additional cemetery point $partial$. In a certain sense, the three parameters that determine the birth-death process correspond to its killing, reflecting, and jumping behaviors at $infty$ used for the one-point compactification, respectively. In general, providing a clear description of the trajectories of a birth-death process, especially in the pathological case where $|nu|=infty$, is challenging. This paper aims to address this issue by studying the birth-death process using approximation methods. Specifically, we will approximate the birth-death process with simpler birth-death processes that are easier to comprehend. For two typical approximation methods, our main results establish the weak convergence of a sequence of probability measures, which are induced by the approximating processes, on the space of all c`adl`ag functions. This type of convergence is significantly stronger than the convergence of transition matrices typically considered in the theory of continuous-time Markov chains.
{"title":"Approximation of birth-death processes","authors":"Liping Li","doi":"arxiv-2409.05018","DOIUrl":"https://doi.org/arxiv-2409.05018","url":null,"abstract":"The birth-death process is a special type of continuous-time Markov chain\u0000with index set $mathbb{N}$. Its resolvent matrix can be fully characterized by\u0000a set of parameters $(gamma, beta, nu)$, where $gamma$ and $beta$ are\u0000non-negative constants, and $nu$ is a positive measure on $mathbb{N}$. By\u0000employing the Ray-Knight compactification, the birth-death process can be\u0000realized as a c`adl`ag process with strong Markov property on the one-point\u0000compactification space $overline{mathbb{N}}_{partial}$, which includes an\u0000additional cemetery point $partial$. In a certain sense, the three parameters\u0000that determine the birth-death process correspond to its killing, reflecting,\u0000and jumping behaviors at $infty$ used for the one-point compactification,\u0000respectively. In general, providing a clear description of the trajectories of a\u0000birth-death process, especially in the pathological case where $|nu|=infty$,\u0000is challenging. This paper aims to address this issue by studying the\u0000birth-death process using approximation methods. Specifically, we will\u0000approximate the birth-death process with simpler birth-death processes that are\u0000easier to comprehend. For two typical approximation methods, our main results\u0000establish the weak convergence of a sequence of probability measures, which are\u0000induced by the approximating processes, on the space of all c`adl`ag\u0000functions. This type of convergence is significantly stronger than the\u0000convergence of transition matrices typically considered in the theory of\u0000continuous-time Markov chains.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212151","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 establish an ordinary as well as a logarithmical convexity of the Moment Generating Function (MGF) for the centered random variable and vector (r.v.) satisfying the Kramer's condition. Our considerations are based on the theory of the so-called Grand Lebesgue Spaces.
{"title":"Ordinary and logarithmical convexity of moment generating function","authors":"M. R. Formica, E. Ostrovsky, L. Sirota","doi":"arxiv-2409.05085","DOIUrl":"https://doi.org/arxiv-2409.05085","url":null,"abstract":"We establish an ordinary as well as a logarithmical convexity of the Moment\u0000Generating Function (MGF) for the centered random variable and vector (r.v.)\u0000satisfying the Kramer's condition. Our considerations are based on the theory of the so-called Grand Lebesgue\u0000Spaces.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212059","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}
This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.
{"title":"Limit theorems under heavy-tailed scenario in the age dependent random connection models","authors":"Christian Hirsch, Takashi Owada","doi":"arxiv-2409.05226","DOIUrl":"https://doi.org/arxiv-2409.05226","url":null,"abstract":"This paper considers limit theorems associated with subgraph counts in the\u0000age-dependent random connection model. First, we identify regimes where the\u0000count of sub-trees converges weakly to a stable random variable under suitable\u0000assumptions on the shape of trees. The proof relies on an intermediate result\u0000on weak convergence of associated point processes towards a Poisson point\u0000process. Additionally, we prove the same type of results for the clique counts.\u0000Here, a crucial ingredient includes the expectation asymptotics for clique\u0000counts, which itself is a result of independent interest.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212058","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}
A recent paper by Bhatia, Chin, Mani, and Mossel (2024) defined stochastic processes which aim to model the game of war for two players for $n$ cards. They showed that these models are equivalent to gambler's ruin and therefore have expected termination time of $Theta(n^2)$. In this paper, we generalize these model to any number of players $m$. We prove for the game with $m$ players is equivalent to a sticky random walk on an $m$-simplex. We show that this implies that the expected termination time is $O(n^2)$. We further provide a lower bound of $Omegaleft(frac{n^2}{m^2}right)$. We conjecture that when $m$ divides $n$, and $n > m$ the termination time or the war game and the absorption times of the sticky random walk are in fact $Theta(n^2)$ uniformly in $m$.
{"title":"On the expected absorption times of sticky random walks and multiple players war games","authors":"Axel Adjei, Elchanan Mossel","doi":"arxiv-2409.05201","DOIUrl":"https://doi.org/arxiv-2409.05201","url":null,"abstract":"A recent paper by Bhatia, Chin, Mani, and Mossel (2024) defined stochastic\u0000processes which aim to model the game of war for two players for $n$ cards.\u0000They showed that these models are equivalent to gambler's ruin and therefore\u0000have expected termination time of $Theta(n^2)$. In this paper, we generalize\u0000these model to any number of players $m$. We prove for the game with $m$\u0000players is equivalent to a sticky random walk on an $m$-simplex. We show that\u0000this implies that the expected termination time is $O(n^2)$. We further provide\u0000a lower bound of $Omegaleft(frac{n^2}{m^2}right)$. We conjecture that when\u0000$m$ divides $n$, and $n > m$ the termination time or the war game and the\u0000absorption times of the sticky random walk are in fact $Theta(n^2)$ uniformly\u0000in $m$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212150","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 matrix integrals of the form $$int_{mathrm{USp(2n)}}prod_{j=1}^kmathrm{tr}(U^j)^{a_j}mathrm d U,$$ where $a_1,ldots,a_r$ are natural numbers and integration is with respect to the Haar probability measure. We obtain a compact formula (the number of terms depends only on $sum a_j$ and not on $n,k$) for the above integral in the non-Gaussian range $sum_{j=1}^kja_jle 4n+1$. This extends results of Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral valid in the (Gaussian) range $sum_{j=1}^kja_jle n$ and $sum_{j=1}^kja_jle 2n+1$ respectively. We derive our formula using the connection between random symplectic matrices and hyperelliptic $L$-functions over finite fields, given by an equidistribution result of Katz and Sarnak, and an evaluation of a certain multiple character sum over the function field $mathbb F_q(x)$. We apply our formula to study the linear statistics of eigenvalues of random unitary symplectic matrices in a narrow bandwidth sampling regime.
{"title":"Moments of traces of random symplectic matrices and hyperelliptic $L$-functions","authors":"Alexei Entin, Noam Pirani","doi":"arxiv-2409.04844","DOIUrl":"https://doi.org/arxiv-2409.04844","url":null,"abstract":"We study matrix integrals of the form\u0000$$int_{mathrm{USp(2n)}}prod_{j=1}^kmathrm{tr}(U^j)^{a_j}mathrm d U,$$\u0000where $a_1,ldots,a_r$ are natural numbers and integration is with respect to\u0000the Haar probability measure. We obtain a compact formula (the number of terms\u0000depends only on $sum a_j$ and not on $n,k$) for the above integral in the\u0000non-Gaussian range $sum_{j=1}^kja_jle 4n+1$. This extends results of\u0000Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral\u0000valid in the (Gaussian) range $sum_{j=1}^kja_jle n$ and $sum_{j=1}^kja_jle\u00002n+1$ respectively. We derive our formula using the connection between random\u0000symplectic matrices and hyperelliptic $L$-functions over finite fields, given\u0000by an equidistribution result of Katz and Sarnak, and an evaluation of a\u0000certain multiple character sum over the function field $mathbb F_q(x)$. We\u0000apply our formula to study the linear statistics of eigenvalues of random\u0000unitary symplectic matrices in a narrow bandwidth sampling regime.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212153","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 this paper, we present several path properties, simulations, inferences, and generalizations of the weighted sub-fractional Brownian motion. A primary focus is on the derivation of the covariance function $R_{f,b}(s,t)$ for the weighted sub-fractional Brownian motion, defined as: begin{equation*} R_{f,b}(s,t) = frac{1}{1-b} int_{0}^{s wedge t} f(r) left[(s-r)^{b} + (t-r)^{b} - (t+s-2r)^{b}right] dr, end{equation*} where $f:mathbb{R}_{+} to mathbb{R}_{+}$ is a measurable function and $bin [0,1)cup(1,2]$. This covariance function $R_{f,b}(s,t)$ is used to define the centered Gaussian process $zeta_{t,f,b}$, which is the weighted sub-fractional Brownian motion. Furthermore, if there is a positive constant $c$ and $a in (-1,infty)$ such that $0 leq f(u) leq c u^{a}$ on $[0,T]$ for some $T>0$. Then, for $b in (0,1)$, $zeta_{t,f,b}$ exhibits infinite variation and zero quadratic variation, making it a non-semi-martingale. On the other hand, for $b in (1,2]$, $zeta_{t,f,b}$ is a continuous process of finite variation and thus a semi-martingale and for $b=0$ the process $zeta_{t,f,0}$ is a square integrable continuous martingale. We also provide inferential studies using maximum likelihood estimation and perform simulations comparing various numerical methods for their efficiency in computing the finite-dimensional distributions of $zeta_{t,f,b}$. Additionally, we extend the weighted sub-fractional Brownian motion to $mathbb{R}^d$ by defining new covariance structures for measurable, bounded sets in $mathbb{R}^d$. Finally, we define a stochastic integral with respect to $zeta_{t,f,b}$ and introduce both the weighted sub-fractional Ornstein-Uhlenbeck process and the geometric weighted sub-fractional Brownian motion.
{"title":"Weighted Sub-fractional Brownian Motion Process: Properties and Generalizations","authors":"Ramirez-Gonzalez Jose Hermenegildo, Sun Ying","doi":"arxiv-2409.04798","DOIUrl":"https://doi.org/arxiv-2409.04798","url":null,"abstract":"In this paper, we present several path properties, simulations, inferences,\u0000and generalizations of the weighted sub-fractional Brownian motion. A primary\u0000focus is on the derivation of the covariance function $R_{f,b}(s,t)$ for the\u0000weighted sub-fractional Brownian motion, defined as: begin{equation*}\u0000R_{f,b}(s,t) = frac{1}{1-b} int_{0}^{s wedge t} f(r) left[(s-r)^{b} +\u0000(t-r)^{b} - (t+s-2r)^{b}right] dr, end{equation*} where $f:mathbb{R}_{+} to\u0000mathbb{R}_{+}$ is a measurable function and $bin [0,1)cup(1,2]$. This\u0000covariance function $R_{f,b}(s,t)$ is used to define the centered Gaussian\u0000process $zeta_{t,f,b}$, which is the weighted sub-fractional Brownian motion.\u0000Furthermore, if there is a positive constant $c$ and $a in (-1,infty)$ such\u0000that $0 leq f(u) leq c u^{a}$ on $[0,T]$ for some $T>0$. Then, for $b in\u0000(0,1)$, $zeta_{t,f,b}$ exhibits infinite variation and zero quadratic\u0000variation, making it a non-semi-martingale. On the other hand, for $b in\u0000(1,2]$, $zeta_{t,f,b}$ is a continuous process of finite variation and thus a\u0000semi-martingale and for $b=0$ the process $zeta_{t,f,0}$ is a square\u0000integrable continuous martingale. We also provide inferential studies using\u0000maximum likelihood estimation and perform simulations comparing various\u0000numerical methods for their efficiency in computing the finite-dimensional\u0000distributions of $zeta_{t,f,b}$. Additionally, we extend the weighted\u0000sub-fractional Brownian motion to $mathbb{R}^d$ by defining new covariance\u0000structures for measurable, bounded sets in $mathbb{R}^d$. Finally, we define a\u0000stochastic integral with respect to $zeta_{t,f,b}$ and introduce both the\u0000weighted sub-fractional Ornstein-Uhlenbeck process and the geometric weighted\u0000sub-fractional Brownian motion.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212154","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}
Motivated by the $(q,gamma)$-cumulants, introduced by Xu [arXiv:2303.13812] to study $beta$-deformed singular values of random matrices, we define the $(n,d)$-rectangular cumulants for polynomials of degree $d$ and prove several moment-cumulant formulas by elementary algebraic manipulations; the proof naturally leads to quantum analogues of the formulas. We further show that the $(n,d)$-rectangular cumulants linearize the $(n,d)$-rectangular convolution from Finite Free Probability and that they converge to the $q$-rectangular free cumulants from Free Probability in the regime where $dtoinfty$, $1+n/dto qin[1,infty)$. As an application, we employ our formulas to study limits of symmetric empirical root distributions of sequences of polynomials with nonnegative roots. One of our results is akin to a theorem of Kabluchko [arXiv:2203.05533] and shows that applying the operator $exp(-frac{s^2}{n}x^{-n}D_xx^{n+1}D_x)$, where $s>0$, asymptotically amounts to taking the rectangular free convolution with the rectangular Gaussian distribution of variance $qs^2/(q-1)$.
{"title":"Cumulants in rectangular finite free probability and beta-deformed singular values","authors":"Cesar Cuenca","doi":"arxiv-2409.04305","DOIUrl":"https://doi.org/arxiv-2409.04305","url":null,"abstract":"Motivated by the $(q,gamma)$-cumulants, introduced by Xu [arXiv:2303.13812]\u0000to study $beta$-deformed singular values of random matrices, we define the\u0000$(n,d)$-rectangular cumulants for polynomials of degree $d$ and prove several\u0000moment-cumulant formulas by elementary algebraic manipulations; the proof\u0000naturally leads to quantum analogues of the formulas. We further show that the\u0000$(n,d)$-rectangular cumulants linearize the $(n,d)$-rectangular convolution\u0000from Finite Free Probability and that they converge to the $q$-rectangular free\u0000cumulants from Free Probability in the regime where $dtoinfty$, $1+n/dto\u0000qin[1,infty)$. As an application, we employ our formulas to study limits of\u0000symmetric empirical root distributions of sequences of polynomials with\u0000nonnegative roots. One of our results is akin to a theorem of Kabluchko\u0000[arXiv:2203.05533] and shows that applying the operator\u0000$exp(-frac{s^2}{n}x^{-n}D_xx^{n+1}D_x)$, where $s>0$, asymptotically amounts\u0000to taking the rectangular free convolution with the rectangular Gaussian\u0000distribution of variance $qs^2/(q-1)$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212168","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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