. We consider the two-star model, a family of exponential random graphs indexed by two real parameters, h and α , that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, if α, h ≥ 0 , we derive first and second order correlation inequalities and then prove the so-called GHS inequality. As a consequence, under the above conditions on the parameters, the average edge density turns out to be an increasing and concave function of the parameter h , at any fixed size of the graph. Some of our results can be extended to more general classes of exponential random graphs.
{"title":"The GHS and other correlation inequalities for the two-star model","authors":"A. Bianchi, Francesca Collet, Elena Magnanini","doi":"10.30757/ALEA.v19-64","DOIUrl":"https://doi.org/10.30757/ALEA.v19-64","url":null,"abstract":". We consider the two-star model, a family of exponential random graphs indexed by two real parameters, h and α , that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, if α, h ≥ 0 , we derive first and second order correlation inequalities and then prove the so-called GHS inequality. As a consequence, under the above conditions on the parameters, the average edge density turns out to be an increasing and concave function of the parameter h , at any fixed size of the graph. Some of our results can be extended to more general classes of exponential random graphs.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47560457","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 large deviations principle for the empirical measure of the one dimensional symmetric simple exclusion process in contact with reservoirs. The dynamics of the reservoirs is slowed down with respect to the dynamics of the system, that is, the rate at which the system exchanges particles with the boundary reservoirs is of order $n^{-theta}$, where $n$ is number of sites in the system, $theta$ is a non negative parameter, and the system is taken in the diffusive time scaling. Two regimes are studied here, the subcritical $thetain(0,1)$ whose hydrodynamic equation is the heat equation with Dirichlet boundary conditions and the supercritical $thetain(1,+infty)$ whose hydrodynamic equation is the heat equation with Neumann boundary conditions. In the subcritical case $thetain(0,1)$, the rate function that we obtain matches the rate function corresponding to the case $theta=0$ which was derived on previous works (see cite{blm,flm}), but the challenges we faced here are much trickier. In the supercritical case $thetain(1,+infty)$, the rate function is equal to infinity outside the set of trajectories which preserve the total mass, meaning that, despite the discrete system exchanges particles with the reservoirs, this phenomena has super-exponentially small probability in the diffusive scaling limit.
{"title":"Large Deviations for the SSEP with slow boundary: the non-critical case","authors":"T. Franco, P. Gonccalves, A. Neumann","doi":"10.30757/alea.v20-13","DOIUrl":"https://doi.org/10.30757/alea.v20-13","url":null,"abstract":"We prove a large deviations principle for the empirical measure of the one dimensional symmetric simple exclusion process in contact with reservoirs. The dynamics of the reservoirs is slowed down with respect to the dynamics of the system, that is, the rate at which the system exchanges particles with the boundary reservoirs is of order $n^{-theta}$, where $n$ is number of sites in the system, $theta$ is a non negative parameter, and the system is taken in the diffusive time scaling. Two regimes are studied here, the subcritical $thetain(0,1)$ whose hydrodynamic equation is the heat equation with Dirichlet boundary conditions and the supercritical $thetain(1,+infty)$ whose hydrodynamic equation is the heat equation with Neumann boundary conditions. In the subcritical case $thetain(0,1)$, the rate function that we obtain matches the rate function corresponding to the case $theta=0$ which was derived on previous works (see cite{blm,flm}), but the challenges we faced here are much trickier. In the supercritical case $thetain(1,+infty)$, the rate function is equal to infinity outside the set of trajectories which preserve the total mass, meaning that, despite the discrete system exchanges particles with the reservoirs, this phenomena has super-exponentially small probability in the diffusive scaling limit.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45737754","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}
Jorge Ignacio Gonz'alez C'azares, Aleksandar Mijatovi'c
We establish a novel characterisation of the law of the convex minorant of any L'evy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of L'evy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of L'evy processes, circumventing local time and excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities and the Wiener-Hopf factorisation, as well as a novel factorisation identity.
{"title":"Convex minorants and the fluctuation theory of Lévy processes","authors":"Jorge Ignacio Gonz'alez C'azares, Aleksandar Mijatovi'c","doi":"10.30757/ALEA.v19-39","DOIUrl":"https://doi.org/10.30757/ALEA.v19-39","url":null,"abstract":"We establish a novel characterisation of the law of the convex minorant of any L'evy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of L'evy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of L'evy processes, circumventing local time and excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities and the Wiener-Hopf factorisation, as well as a novel factorisation identity.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46595340","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 study the degree distribution of a randomly chosen vertex in a duplication–divergence graph, under a variety of different generalizations of the basic model of Bhan et al. (2002) and Vázquez et al. (2003). We pay particular attention to what happens when a non-trivial proportion of the vertices have large degrees, establishing a central limit theorem for the logarithm of the degree distribution. Our approach, as in Jordan (2018) and Hermann and Pfaffelhuber (2021), relies heavily on the analysis of related birth–catastrophe processes, and couplings are used to show that a number of different formulations of the process have asymptotically similar expected degree distributions.
在Bhan et al.(2002)和Vázquez et al.(2003)的基本模型的各种不同推广下，我们研究了重复发散图中随机选择顶点的度分布。我们特别关注当一个非平凡比例的顶点具有较大的度时会发生什么，为度分布的对数建立了一个中心极限定理。我们的方法，如Jordan(2018)和Hermann和Pfaffelhuber(2021)，在很大程度上依赖于对相关的出生-灾难过程的分析，并使用耦合来表明该过程的许多不同公式具有渐近相似的预期度分布。
{"title":"The expected degree distribution in transient duplication divergence models","authors":"A. Barbour, Tiffany Y. Y. Lo","doi":"10.30757/alea.v19-04","DOIUrl":"https://doi.org/10.30757/alea.v19-04","url":null,"abstract":"We study the degree distribution of a randomly chosen vertex in a duplication–divergence graph, under a variety of different generalizations of the basic model of Bhan et al. (2002) and Vázquez et al. (2003). We pay particular attention to what happens when a non-trivial proportion of the vertices have large degrees, establishing a central limit theorem for the logarithm of the degree distribution. Our approach, as in Jordan (2018) and Hermann and Pfaffelhuber (2021), relies heavily on the analysis of related birth–catastrophe processes, and couplings are used to show that a number of different formulations of the process have asymptotically similar expected degree distributions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45238899","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 consider the superposition of symmetric simple exclusion dynamics speeded-up in time, with spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We show that the mixing time has an exponential lower bound in the system size if the potential of the hydrodynamic equation has two or more local minima. We also apply our estimates to show that the normalized hitting times of rare events converge to a mean one exponential random variable if the potential has a unique minimum. deviation the quasi-potential and solutions to the
{"title":"Exponentially slow mixing and hitting times of rare events for a reaction–diffusion model","authors":"K. Tsunoda","doi":"10.30757/alea.v19-48","DOIUrl":"https://doi.org/10.30757/alea.v19-48","url":null,"abstract":". We consider the superposition of symmetric simple exclusion dynamics speeded-up in time, with spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We show that the mixing time has an exponential lower bound in the system size if the potential of the hydrodynamic equation has two or more local minima. We also apply our estimates to show that the normalized hitting times of rare events converge to a mean one exponential random variable if the potential has a unique minimum. deviation the quasi-potential and solutions to the","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591340","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}
The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout &Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki &Delmas (2020) about the central limit theorem for general additive functionals of bifurcating Markov chains, we give here a moderate deviation principle for additive functionals of bifurcating Markov chains when the functions depend on one variable. This work is done under the uniform geometric ergodicity and the uniform ergodic property based on the second spectral gap assumptions. The proofs of our results are based on martingale decomposition recently developed by Bitseki &Delmas (2020) and on results of Dembo (1996), Djellout (2001) and Puhalski (1997).
{"title":"Moderate deviation principles for bifurcating Markov chains: case of functions dependent of one variable","authors":"S. Valère, Bitseki Penda, Gorgui Gackou","doi":"10.30757/alea.v19-24","DOIUrl":"https://doi.org/10.30757/alea.v19-24","url":null,"abstract":"The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout &Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki &Delmas (2020) about the central limit theorem for general additive functionals of bifurcating Markov chains, we give here a moderate deviation principle for additive functionals of bifurcating Markov chains when the functions depend on one variable. This work is done under the uniform geometric ergodicity and the uniform ergodic property based on the second spectral gap assumptions. The proofs of our results are based on martingale decomposition recently developed by Bitseki &Delmas (2020) and on results of Dembo (1996), Djellout (2001) and Puhalski (1997).","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49042150","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 introduces a stochastic adaptive dynamics model for the interplay of several crucial traits and mechanisms in bacterial evolution, namely dormancy, horizontal gene transfer (HGT), mutation and competition. In particular, it combines the recent model of Champagnat, M'el'eard and Tran (2021) involving HGT with the model for competition-induced dormancy of Blath and T'obi'as (2020). Our main result is a convergence theorem which describes the evolution of the different traits in the population on a `doubly logarithmic scale' as piece-wise affine functions. Interestingly, even for a relatively small trait space, the limiting process exhibits a non-monotone dependence of the success of the dormancy trait on the dormancy initiation probability. Further, the model establishes a new `approximate coexistence regime' for multiple traits that has not been observed in previous literature.
{"title":"A stochastic adaptive dynamics model for bacterial populations with mutation, dormancy and transfer","authors":"J. Blath, T. Paul, Andr'as T'obi'as","doi":"10.30757/alea.v20-12","DOIUrl":"https://doi.org/10.30757/alea.v20-12","url":null,"abstract":"This paper introduces a stochastic adaptive dynamics model for the interplay of several crucial traits and mechanisms in bacterial evolution, namely dormancy, horizontal gene transfer (HGT), mutation and competition. In particular, it combines the recent model of Champagnat, M'el'eard and Tran (2021) involving HGT with the model for competition-induced dormancy of Blath and T'obi'as (2020). Our main result is a convergence theorem which describes the evolution of the different traits in the population on a `doubly logarithmic scale' as piece-wise affine functions. Interestingly, even for a relatively small trait space, the limiting process exhibits a non-monotone dependence of the success of the dormancy trait on the dormancy initiation probability. Further, the model establishes a new `approximate coexistence regime' for multiple traits that has not been observed in previous literature.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47307384","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 study the directed polymer model in a bounded environment with bond disorder and show that, in the interior of the weak disorder phase, weak disorder continues to hold upon perturbation by a small bias. Using this stability result, we give a new proof for the central limit theorem (CLT) in probability for the directed polymer model in the interior of the weak disorder phase. We also show that the large deviation rate function agrees with that of the underlying random walk. For the Brownian polymer model, we improve the convergence in the CLT to almost sure convergence in the whole weak disorder phase. The main technical tools are a new moment bound from cite{J21_1} and a quantitative comparison between the associated martingales at different inverse temperatures.
{"title":"Stability of weak disorder phase for directed polymer with applications to limit theorems","authors":"S. Junk","doi":"10.30757/ALEA.v20-31","DOIUrl":"https://doi.org/10.30757/ALEA.v20-31","url":null,"abstract":"We study the directed polymer model in a bounded environment with bond disorder and show that, in the interior of the weak disorder phase, weak disorder continues to hold upon perturbation by a small bias. Using this stability result, we give a new proof for the central limit theorem (CLT) in probability for the directed polymer model in the interior of the weak disorder phase. We also show that the large deviation rate function agrees with that of the underlying random walk. For the Brownian polymer model, we improve the convergence in the CLT to almost sure convergence in the whole weak disorder phase. The main technical tools are a new moment bound from cite{J21_1} and a quantitative comparison between the associated martingales at different inverse temperatures.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48244668","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 analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $log$-H"older continuous.
我们分析了几种无序统计力学模型分析中出现的2 × 2矩阵序列乘积的上李雅普诺夫指数:例如，这些矩阵是具有随机外场的最近邻伊辛链的传递矩阵，而这个伊辛链的自由能密度就是我们所考虑的李雅普诺夫指数。当外场为中心时，我们得到了该指数在大相互作用极限下的尖锐行为:这种平衡情况在许多方面都是至关重要的。从数学的角度，我们精确地确定了一个接近于上下李雅普诺夫指数重合的对角随机矩阵的二维随机矩阵的乘积的上李雅普诺夫指数的行为。特别是,李雅普诺夫指数只是 log - h “老美元连续的。
{"title":"Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case","authors":"G. Giacomin, R. L. Greenblatt","doi":"10.30757/ALEA.v19-27","DOIUrl":"https://doi.org/10.30757/ALEA.v19-27","url":null,"abstract":"We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $log$-H\"older continuous.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47582416","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}
The $beta$-Delaunay tessellation in $mathbb{R}^{d-1}$ is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a $beta$-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in $mathbb{R}^{d-1}$. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional $beta$-Delaunay tessellation is analysed, as $dtoinfty$. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived.
{"title":"The β-Delaunay tessellation III: Kendall’s problem and limit theorems in high dimensions","authors":"A. Gusakova, Z. Kabluchko, Christoph Thale","doi":"10.30757/ALEA.v19-02","DOIUrl":"https://doi.org/10.30757/ALEA.v19-02","url":null,"abstract":"The $beta$-Delaunay tessellation in $mathbb{R}^{d-1}$ is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a $beta$-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in $mathbb{R}^{d-1}$. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional $beta$-Delaunay tessellation is analysed, as $dtoinfty$. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49369872","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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