Driven by the explosion of data and the impact of real-world networks, a wide array of mathematical models have been proposed to understand the structure and evolution of such systems, especially in the temporal context. Recent advances in areas such as distributed cyber-security and social networks have motivated the development of probabilistic models of evolution where individuals have only partial information on the state of the network when deciding on their actions. This paper aims to understand models incorporating emph{network delay}, where new individuals have information on a time-delayed snapshot of the system. We consider the setting where one has macroscopic delays, that is, the ``information'' available to new individuals is the structure of the network at a past time, which scales proportionally with the current time and vertices connect using standard preferential attachment type dynamics. We obtain the local weak limit for the network as its size grows and connect it to a novel continuous-time branching process where the associated point process of reproductions emph{has memory} of the entire past. A more tractable `dual description' of this branching process using an `edge copying mechanism' is used to obtain degree distribution asymptotics as well as necessary and sufficient conditions for condensation, where the mass of the degree distribution ``escapes to infinity''. We conclude by studying the impact of the delay distribution on macroscopic functionals such as the root degree.
{"title":"Network evolution with Macroscopic Delays: asymptotics and condensation","authors":"Sayan Banerjee, Shankar Bhamidi, Partha Dey, Akshay Sakanaveeti","doi":"arxiv-2409.06048","DOIUrl":"https://doi.org/arxiv-2409.06048","url":null,"abstract":"Driven by the explosion of data and the impact of real-world networks, a wide\u0000array of mathematical models have been proposed to understand the structure and\u0000evolution of such systems, especially in the temporal context. Recent advances\u0000in areas such as distributed cyber-security and social networks have motivated\u0000the development of probabilistic models of evolution where individuals have\u0000only partial information on the state of the network when deciding on their\u0000actions. This paper aims to understand models incorporating emph{network\u0000delay}, where new individuals have information on a time-delayed snapshot of\u0000the system. We consider the setting where one has macroscopic delays, that is,\u0000the ``information'' available to new individuals is the structure of the\u0000network at a past time, which scales proportionally with the current time and\u0000vertices connect using standard preferential attachment type dynamics. We\u0000obtain the local weak limit for the network as its size grows and connect it to\u0000a novel continuous-time branching process where the associated point process of\u0000reproductions emph{has memory} of the entire past. A more tractable `dual\u0000description' of this branching process using an `edge copying mechanism' is\u0000used to obtain degree distribution asymptotics as well as necessary and\u0000sufficient conditions for condensation, where the mass of the degree\u0000distribution ``escapes to infinity''. We conclude by studying the impact of the\u0000delay distribution on macroscopic functionals such as the root degree.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212049","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}
Arijit Chakrabarty, Rajat Subhra Hazra, Moumanti Podder
This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that the largest eigenvalue, after centering, converges in distribution to normal with an explicitly defined mean and variance. This result generalizes known findings for Wigner matrices with independent entries.
{"title":"Largest eigenvalue of positive mean Gaussian matrices","authors":"Arijit Chakrabarty, Rajat Subhra Hazra, Moumanti Podder","doi":"arxiv-2409.05858","DOIUrl":"https://doi.org/arxiv-2409.05858","url":null,"abstract":"This short note studies the fluctuations of the largest eigenvalue of\u0000symmetric random matrices with correlated Gaussian entries having positive\u0000mean. Under the assumption that the covariance kernel is absolutely summable,\u0000it is proved that the largest eigenvalue, after centering, converges in\u0000distribution to normal with an explicitly defined mean and variance. This\u0000result generalizes known findings for Wigner matrices with independent entries.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212050","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 correct errors that appear throughout "The vicious neighbour problem" by Tao and Wu. We seek to solve the following problem. Suppose Nnerds are distributed uniformly at random in a square region. At 3:14pm, every nerd simultaneously snipes their nearest neighbor. What is the expected proportion $P_N$ of nerds who are left unscathed in the limit as $Ntoinfty$?
我们纠正了陶和吴在《恶性相邻问题》一文中出现的错误。我们试图解决以下问题。假设 N 个书呆子均匀地随机分布在一个正方形区域中。下午 3 点 14 分，每个书呆子都同时 "攻击 "了他们最近的邻居。在$Ntoinfty$的极限中，毫发无损的书呆子的预期比例$P_N$是多少？
{"title":"The nerd snipers problem","authors":"Boris Alexeev, Dustin Mixon","doi":"arxiv-2409.06068","DOIUrl":"https://doi.org/arxiv-2409.06068","url":null,"abstract":"We correct errors that appear throughout \"The vicious neighbour problem\" by\u0000Tao and Wu. We seek to solve the following problem. Suppose Nnerds are distributed\u0000uniformly at random in a square region. At 3:14pm, every nerd simultaneously\u0000snipes their nearest neighbor. What is the expected proportion $P_N$ of nerds\u0000who are left unscathed in the limit as $Ntoinfty$?","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"264 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212046","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 investigate the convergence towards equilibrium of the noisy voter model, evolving in the complete graph with n vertices. The noisy voter model is a version of the voter model, on which individuals change their opinions randomly due to external noise. Specifically, we determine the profile of convergence, in Kantorovich distance (also known as 1-Wasserstein distance), which corresponds to the Kantorovich distance between the marginals of a Wright-Fisher diffusion and its stationary measure. In particular, we demonstrate that the model does not exhibit cut-off under natural noise intensity conditions. In addition, we study the time the model needs to forget the initial location of particles, which we interpret as the Kantorovich distance between the laws of the model with particles in fixed initial positions and in positions chosen uniformly at random. We call this process thermalization and we show that thermalization does exhibit a cut-off profile. Our approach relies on Stein's method and analytical tools from PDE theory, which may be of independent interest for the quantitative study of observables of Markov chains.
我们研究了在有 n 个顶点的完整图中演化的噪声选民模型向均衡收敛的问题。噪声投票者模型是投票者模型的反转，在该模型中，个体会因外部噪声而随机改变自己的观点。具体来说，我们确定了收敛曲线的康托洛维奇距离（也称为 1-Wasserstein 距离），它对应于赖特-费舍扩散的边际与其静态度量之间的康托洛维奇距离。我们特别证明，该模型在自然噪声强度条件下不会出现截断现象。此外，我们还研究了模型遗忘粒子初始位置所需的时间，我们将其解释为粒子处于固定初始位置时模型规律与均匀随机选择位置时模型规律之间的康托洛维奇距离。我们的方法依赖于斯坦因方法和 PDE 理论的分析工具，这些工具对于马尔可夫链观测值的定量研究可能具有独立的意义。
{"title":"Thermalization And Convergence To Equilibrium Of The Noisy Voter Model","authors":"Enzo Aljovin, Milton Jara, Yangrui Xiang","doi":"arxiv-2409.05722","DOIUrl":"https://doi.org/arxiv-2409.05722","url":null,"abstract":"We investigate the convergence towards equilibrium of the noisy voter model,\u0000evolving in the complete graph with n vertices. The noisy voter model is a\u0000version of the voter model, on which individuals change their opinions randomly\u0000due to external noise. Specifically, we determine the profile of convergence,\u0000in Kantorovich distance (also known as 1-Wasserstein distance), which\u0000corresponds to the Kantorovich distance between the marginals of a\u0000Wright-Fisher diffusion and its stationary measure. In particular, we\u0000demonstrate that the model does not exhibit cut-off under natural noise\u0000intensity conditions. In addition, we study the time the model needs to forget\u0000the initial location of particles, which we interpret as the Kantorovich\u0000distance between the laws of the model with particles in fixed initial\u0000positions and in positions chosen uniformly at random. We call this process\u0000thermalization and we show that thermalization does exhibit a cut-off profile.\u0000Our approach relies on Stein's method and analytical tools from PDE theory,\u0000which may be of independent interest for the quantitative study of observables\u0000of Markov chains.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212053","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 spatial logistic branching process is a population dynamics model in which particles move on a lattice according to independent simple symmetric random walks, each particle splits into a random number of individuals at rate one, and pairs of particles at the same location compete at rate c. We consider the weak competition regime in which c tends to zero, corresponding to a local carrying capacity tending to infinity like 1/c. We show that the hydrodynamic limit of the spatial logistic branching process is given by the Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck process with deterministic but heterogeneous coefficients. The proofs rely on an adaptation of the method of v-functions developed in Boldrighini et al. 1992. An intermediate result of independent interest shows how the tail of the offspring distribution and the precise regime in which c tends to zero affect the convergence rate of the expected population size of the spatial logistic branching process to the hydrodynamic limit.
空间对数分支过程是一个种群动力学模型，其中粒子按照独立的简单对称随机行走在晶格上移动，每个粒子以速率1分裂成随机数目的个体，同一位置的粒子对以速率c竞争。我们考虑了弱竞争机制，其中c趋于零，对应于趋于无穷大的局部承载能力，如1/c。我们证明，空间逻辑分支过程的流体力学极限是由 Fisher-Kolmogorov-Petrovsky-Piskunov 方程给出的。然后，我们证明了它的非平衡波动收敛于一个具有确定但异质系数的广义奥恩斯坦-乌伦贝克过程。证明依赖于对 Boldrighini 等人 1992 年提出的 v 函数方法的调整。一个令人感兴趣的中间结果显示了后代分布的尾部和 c 趋于零的精确机制如何影响空间对数分支过程的预期种群数量向流体力学极限的收敛速度。
{"title":"Non-Equilibrium Fluctuations for a Spatial Logistic Branching Process with Weak Competition","authors":"Thomas Tendron","doi":"arxiv-2409.05269","DOIUrl":"https://doi.org/arxiv-2409.05269","url":null,"abstract":"The spatial logistic branching process is a population dynamics model in\u0000which particles move on a lattice according to independent simple symmetric\u0000random walks, each particle splits into a random number of individuals at rate\u0000one, and pairs of particles at the same location compete at rate c. We consider\u0000the weak competition regime in which c tends to zero, corresponding to a local\u0000carrying capacity tending to infinity like 1/c. We show that the hydrodynamic\u0000limit of the spatial logistic branching process is given by the\u0000Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its\u0000non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck\u0000process with deterministic but heterogeneous coefficients. The proofs rely on\u0000an adaptation of the method of v-functions developed in Boldrighini et al.\u00001992. An intermediate result of independent interest shows how the tail of the\u0000offspring distribution and the precise regime in which c tends to zero affect\u0000the convergence rate of the expected population size of the spatial logistic\u0000branching process to the hydrodynamic limit.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"91 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212057","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 emptiness formation probability, along with various representations for nonlocal correlation functions, of the 20-vertex model. In doing so, we leverage previous arguments for representations of nonlocal correlation functions for the 6-vertex model, under domain-wall boundary conditions, due to Colomo, Di Giulio, and Pronko, in addition to the inhomogeneous, and homogeneous, determinantal representations for the 20-vertex partition function due to Di Francesco, also under domain-wall boundary conditions. By taking a product of row configuration probabilities, we obtain a desired contour integral representation for nonlocal correlations from a determinantal representation. Finally, a counterpart of the emptiness formation probability is introduced for the 20-vertex model.
{"title":"The emptiness formation probability, and representations for nonlocal correlation functions, of the 20-vertex model","authors":"Pete Rigas","doi":"arxiv-2409.05309","DOIUrl":"https://doi.org/arxiv-2409.05309","url":null,"abstract":"We study the emptiness formation probability, along with various\u0000representations for nonlocal correlation functions, of the 20-vertex model. In\u0000doing so, we leverage previous arguments for representations of nonlocal\u0000correlation functions for the 6-vertex model, under domain-wall boundary\u0000conditions, due to Colomo, Di Giulio, and Pronko, in addition to the\u0000inhomogeneous, and homogeneous, determinantal representations for the 20-vertex\u0000partition function due to Di Francesco, also under domain-wall boundary\u0000conditions. By taking a product of row configuration probabilities, we obtain a\u0000desired contour integral representation for nonlocal correlations from a\u0000determinantal representation. Finally, a counterpart of the emptiness formation\u0000probability is introduced for the 20-vertex model.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212157","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 refinement of the multinomial distribution is presented where the number of inversions in the sequence of outcomes is tallied. This refinement of the multinomial distribution is its joint distribution with the number of inversions in the accompanying experiment. An application of this additional information is described in which the number of inversions acts as a proxy measure of homogeneity (or lack thereof) in the sequence of outcomes.
{"title":"A refinement of the multinomial distribution with application","authors":"Andrew V. Sills","doi":"arxiv-2409.05788","DOIUrl":"https://doi.org/arxiv-2409.05788","url":null,"abstract":"A refinement of the multinomial distribution is presented where the number of\u0000inversions in the sequence of outcomes is tallied. This refinement of the\u0000multinomial distribution is its joint distribution with the number of\u0000inversions in the accompanying experiment. An application of this additional\u0000information is described in which the number of inversions acts as a proxy\u0000measure of homogeneity (or lack thereof) in the sequence of outcomes.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212051","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 note, we consider a sample covariance matrix of the form $$M_{n}=sum_{alpha=1}^m tau_alpha {mathbf{y}}_{alpha}^{(1)} otimes {mathbf{y}}_{alpha}^{(2)}({mathbf{y}}_{alpha}^{(1)} otimes {mathbf{y}}_{alpha}^{(2)})^T,$$ where $(mathbf{y}_{alpha}^{(1)},, {mathbf{y}}_{alpha}^{(2)})_{alpha}$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and $tau_alpha in mathbb{R}_+ $. We show that as $m, n to infty$, $m/n^2to c>0$, the centralized traces of the resolvents, $mathrm{Tr}(M_n-zI_n)^{-1}-mathbf{E}mathrm{Tr}(M_n-zI_n)^{-1}$, $Im zge eta_0>0$, converge in distribution to a two-dimensional Gaussian random variable with zero mean and a certain covariance matrix. This work is a continuation of Dembczak-Ko{l}odziejczyk and Lytova (2023), and Lytova (2018).
{"title":"A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices","authors":"Alicja Dembczak-Kołodziejczyk","doi":"arxiv-2409.06007","DOIUrl":"https://doi.org/arxiv-2409.06007","url":null,"abstract":"In this note, we consider a sample covariance matrix of the form\u0000$$M_{n}=sum_{alpha=1}^m tau_alpha {mathbf{y}}_{alpha}^{(1)} otimes\u0000{mathbf{y}}_{alpha}^{(2)}({mathbf{y}}_{alpha}^{(1)} otimes\u0000{mathbf{y}}_{alpha}^{(2)})^T,$$ where $(mathbf{y}_{alpha}^{(1)},,\u0000{mathbf{y}}_{alpha}^{(2)})_{alpha}$ are independent vectors uniformly\u0000distributed on the unit sphere $S^{n-1}$ and $tau_alpha in mathbb{R}_+ $.\u0000We show that as $m, n to infty$, $m/n^2to c>0$, the centralized traces of\u0000the resolvents,\u0000$mathrm{Tr}(M_n-zI_n)^{-1}-mathbf{E}mathrm{Tr}(M_n-zI_n)^{-1}$, $Im zge\u0000eta_0>0$, converge in distribution to a two-dimensional Gaussian random\u0000variable with zero mean and a certain covariance matrix. This work is a\u0000continuation of Dembczak-Ko{l}odziejczyk and Lytova (2023), and Lytova (2018).","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212044","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 entropic repulsion of the low temperature 3D Ising and Potts interface in an $ntimes n times n$ box with blue boundary conditions on its bottom face (the hard floor), and red boundary conditions on its other five faces. For Ising, Frohlich and Pfister proved in 1987 that the typical interface height above the origin diverges (non-quantitatively), via correlation inequalities special to the Ising model; no such result was known for Potts. We show for both the Ising and Potts models that the entropic repulsion fully overcomes the potentially attractive interaction with the floor, and obtain a logarithmically diverging lower bound on the typical interface height. This is complemented by a conjecturally sharp upper bound of $lfloor xi^{-1}log nrfloor$ where $xi$ is the rate function for a point-to-plane non-red connection under the infinite volume red measure. The proof goes through a coupled random-cluster interface to overcome the potential attractive interaction with the boundary, and a coupled fuzzy Potts model to reduce the upper bound to a simpler setting where the repulsion is attained by conditioning a no-floor interface to lie in the upper half-space.
我们研究了在一个 $ntimes n times n$ 的盒子中低温三维伊辛和波特斯界面的熵斥力，盒子底面（硬地板）为蓝色边界条件，其他五个面为红色边界条件。对于伊辛模型，弗洛里希和普菲斯特在 1987 年证明了原点之上的典型面高度发散（非定量），即伊辛模型所特有的相关不等式；而对于波茨模型，还不知道有这样的结果。我们证明了伊辛模型和波茨模型的熵斥力完全克服了与底面的潜在吸引力相互作用，并得到了典型界面高度的对数发散下限。这又得到了一个猜想中的尖锐上界：$lfloor xi^{-1}log nrfloor$ ，其中$xi$ 是无限体积红色度量下点到平面非红色连接的速率函数。该证明通过一个耦合随机-簇界面来克服与边界的潜在吸引力相互作用，并通过一个耦合模糊波特斯模型将上界还原为一个更简单的设置，即通过将无地板界面设置为位于上半空间来实现斥力。
{"title":"Logarithmic delocalization of low temperature 3D Ising and Potts interfaces above a hard floor","authors":"Joseph Chen, Reza Gheissari, Eyal Lubetzky","doi":"arxiv-2409.06079","DOIUrl":"https://doi.org/arxiv-2409.06079","url":null,"abstract":"We study the entropic repulsion of the low temperature 3D Ising and Potts\u0000interface in an $ntimes n times n$ box with blue boundary conditions on its\u0000bottom face (the hard floor), and red boundary conditions on its other five\u0000faces. For Ising, Frohlich and Pfister proved in 1987 that the typical\u0000interface height above the origin diverges (non-quantitatively), via\u0000correlation inequalities special to the Ising model; no such result was known\u0000for Potts. We show for both the Ising and Potts models that the entropic\u0000repulsion fully overcomes the potentially attractive interaction with the\u0000floor, and obtain a logarithmically diverging lower bound on the typical\u0000interface height. This is complemented by a conjecturally sharp upper bound of\u0000$lfloor xi^{-1}log nrfloor$ where $xi$ is the rate function for a\u0000point-to-plane non-red connection under the infinite volume red measure. The\u0000proof goes through a coupled random-cluster interface to overcome the potential\u0000attractive interaction with the boundary, and a coupled fuzzy Potts model to\u0000reduce the upper bound to a simpler setting where the repulsion is attained by\u0000conditioning a no-floor interface to lie in the upper half-space.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"410 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212042","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 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":"5 1","pages":""},"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}
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