We introduce a new method to evaluate algebraic integrals over the simplex numerically. It improves upon geometric sector decomposition by employing tools from tropical geometry. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.
{"title":"Tropical Monte Carlo quadrature for Feynman integrals","authors":"M. Borinsky","doi":"10.4171/AIHPD/158","DOIUrl":"https://doi.org/10.4171/AIHPD/158","url":null,"abstract":"We introduce a new method to evaluate algebraic integrals over the simplex numerically. It improves upon geometric sector decomposition by employing tools from tropical geometry. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"37 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74178777","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}
For a fixed quadratic polynomial $mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $Ntimes N$ complex Ginibre matrices $X_1^N,dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =mathfrak{p}(X_1^N,dots, X_n^N)$ to the Brown measure of $mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.
对于$n$不可交换变量中的固定二次多项式$mathfrak{p}$，以及$n$独立的$n 乘以n$复Ginibre矩阵$X_1^ n，dots, X_n^ n$，我们建立了$ p ^ n =mathfrak{p} $ (X_1^ n，dots, X_n^ n)$的经验谱分布在$n$自由独立的圆元$c_1，dots, c_n$处的Brown测度在非交换概率空间中的收敛性。证明的主要步骤是获得P^N$伪谱的定量控制。通过众所周知的线性化技巧，这取决于某些矩阵值随机漫步的反集中特性，我们发现，由于与离散标量随机漫步的Littlewood—offford问题中所揭示的算术障碍不同的结构原因，这种特性可能会失败。
{"title":"Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices","authors":"Nicholas A. Cook, A. Guionnet, Jonathan Husson","doi":"10.1214/21-aihp1225","DOIUrl":"https://doi.org/10.1214/21-aihp1225","url":null,"abstract":"For a fixed quadratic polynomial $mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $Ntimes N$ complex Ginibre matrices $X_1^N,dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =mathfrak{p}(X_1^N,dots, X_n^N)$ to the Brown measure of $mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"73 ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72431441","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 relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic domain. Unlike the infinite line case, the limiting one point distribution depends non-trivially on the scaled time parameter. We study several properties of this distribution for the case of the periodic step and flat initial conditions. We show that the distribution changes from a Tracy-Widom distribution in the small time limit to the Gaussian distribution in the large time limit, and also obtain right tail estimate for all time. Furthermore, we establish a connection to integrable differential equations such as the KP equation, coupled systems of mKdV and nonlinear heat equations, and the KdV equation.
{"title":"Limiting one-point distribution of periodic TASEP","authors":"J. Baik, Zhipeng Liu, G. F. D. Silva","doi":"10.1214/21-aihp1171","DOIUrl":"https://doi.org/10.1214/21-aihp1171","url":null,"abstract":"The relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic domain. Unlike the infinite line case, the limiting one point distribution depends non-trivially on the scaled time parameter. We study several properties of this distribution for the case of the periodic step and flat initial conditions. We show that the distribution changes from a Tracy-Widom distribution in the small time limit to the Gaussian distribution in the large time limit, and also obtain right tail estimate for all time. Furthermore, we establish a connection to integrable differential equations such as the KP equation, coupled systems of mKdV and nonlinear heat equations, and the KdV equation.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"72 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85959123","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 the last decade, Hawkes processes have received a lot of attention as good models for functional connectivity in neural spiking networks. In this paper we consider a variant of this process; the age dependent Hawkes process, which incorporates individual post-jump behavior into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean-field type, we study large network limits and establish the propagation of chaos property of the system. du réseau est supprimée ou au moins modifiée. Nous améliorons les résultats de stabilité classiques pour les processus de Hawkes dans ce cadre. En particulier, nous n’avons ni besoin de supposer que les intensités sont bornées, ni d’imposer une condition aux normes Lipschitz des fonctions taux de saut. Lorsque les interactions entre les neurones sont du type champ moyen, nous étudions les limites en grande population et nous démontrons la propriété de propagation du chaos du système.
{"title":"Stability and mean-field limits of age dependent Hawkes processes","authors":"Mads Bonde Raad, Susanne Ditlevsen, E. Löcherbach","doi":"10.1214/19-aihp1023","DOIUrl":"https://doi.org/10.1214/19-aihp1023","url":null,"abstract":". In the last decade, Hawkes processes have received a lot of attention as good models for functional connectivity in neural spiking networks. In this paper we consider a variant of this process; the age dependent Hawkes process, which incorporates individual post-jump behavior into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean-field type, we study large network limits and establish the propagation of chaos property of the system. du réseau est supprimée ou au moins modifiée. Nous améliorons les résultats de stabilité classiques pour les processus de Hawkes dans ce cadre. En particulier, nous n’avons ni besoin de supposer que les intensités sont bornées, ni d’imposer une condition aux normes Lipschitz des fonctions taux de saut. Lorsque les interactions entre les neurones sont du type champ moyen, nous étudions les limites en grande population et nous démontrons la propriété de propagation du chaos du système.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"16 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84375154","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 is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed descriptions of the limit process are given in the case of simple symmetric walks and Gaussian walks. Some open problems are also presented.
{"title":"Extreme order statistics of random walks","authors":"J. Pitman, Wenpin Tang","doi":"10.1214/22-aihp1254","DOIUrl":"https://doi.org/10.1214/22-aihp1254","url":null,"abstract":"This paper is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed descriptions of the limit process are given in the case of simple symmetric walks and Gaussian walks. Some open problems are also presented.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"10 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73498293","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 Hairer-Kelly map has been introduced for establishing a correspondence between geometric and non-geometric rough paths. Recently, a new renormalisation on rough paths has been proposed in (arxiv 1810.12179), built on this map and the Lyons-Victoir extension theorem. In this work, we compare this renormalisation with the existing ones such as BPHZ and the local products renormalisations. We prove that they commute in a certain sense with the Hairer-Kelly map and exhibit an explicit formula in the framework of (arxiv 1810.12179). We also see how the renormalisation behaves in the alternative approach in ( arXiv:1712.01965) for moving from non-geometric to geometric rough paths.
{"title":"Renormalisation from non-geometric to geometric rough paths","authors":"Y. Bruned","doi":"10.1214/21-AIHP1178","DOIUrl":"https://doi.org/10.1214/21-AIHP1178","url":null,"abstract":"The Hairer-Kelly map has been introduced for establishing a correspondence between geometric and non-geometric rough paths. Recently, a new renormalisation on rough paths has been proposed in (arxiv 1810.12179), built on this map and the Lyons-Victoir extension theorem. In this work, we compare this renormalisation with the existing ones such as BPHZ and the local products renormalisations. We prove that they commute in a certain sense with the Hairer-Kelly map and exhibit an explicit formula in the framework of (arxiv 1810.12179). We also see how the renormalisation behaves in the alternative approach in ( arXiv:1712.01965) for moving from non-geometric to geometric rough paths.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"87 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83517792","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 application of formal diffeomorphisms to scalar fields. We give a new proof that interacting tree amplitudes vanish in the resulting theories. Our proof is directly at the diagrammatic level, not appealing to the path integral, and proceeds via a generating function analysis so is more insightful than previous proofs. Along the way we give new combinatorial proofs of some Bell polynomial identities, and we comment on the connection with the combinatorial Legendre transform.
{"title":"Diffeomorphisms of scalar quantum fields via generating functions","authors":"Ali Assem Mahmoud, K. Yeats","doi":"10.4171/aihpd/161","DOIUrl":"https://doi.org/10.4171/aihpd/161","url":null,"abstract":"We study the application of formal diffeomorphisms to scalar fields. We give a new proof that interacting tree amplitudes vanish in the resulting theories. Our proof is directly at the diagrammatic level, not appealing to the path integral, and proceeds via a generating function analysis so is more insightful than previous proofs. Along the way we give new combinatorial proofs of some Bell polynomial identities, and we comment on the connection with the combinatorial Legendre transform.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"71 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75266585","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 article, we study the stochastic wave equation in all dimensions $dleq 3$, driven by a Gaussian noise $dot{W}$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the $p$-th moment of the solution either when the time is large or when $p$ is large. For the critical case, that is the case when $d=3$ and the noise is white, we obtain the exact transition time for the second moment to be finite.
{"title":"Exact asymptotics of the stochastic wave equation with time-independent noise","authors":"R. Balan, Le Chen, Xia Chen","doi":"10.1214/21-aihp1207","DOIUrl":"https://doi.org/10.1214/21-aihp1207","url":null,"abstract":"In this article, we study the stochastic wave equation in all dimensions $dleq 3$, driven by a Gaussian noise $dot{W}$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the $p$-th moment of the solution either when the time is large or when $p$ is large. For the critical case, that is the case when $d=3$ and the noise is white, we obtain the exact transition time for the second moment to be finite.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"240 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85399961","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 recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-$4$ internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure, in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space $mathcal M_{0,4}$. We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.
{"title":"Motives of melonic graphs","authors":"P. Aluffi, M. Marcolli, Waleed Qaisar","doi":"10.4171/aihpd/156","DOIUrl":"https://doi.org/10.4171/aihpd/156","url":null,"abstract":"We investigate recursive relations for the Grothendieck classes of the affine graph hypersurface complements of melonic graphs. We compute these classes explicitly for several families of melonic graphs, focusing on the case of graphs with valence-$4$ internal vertices, relevant to CTKT tensor models. The results hint at a complex and interesting structure, in terms of divisibility relations or nontrivial relations between classes of graphs in different families. Using the recursive relations we prove that the Grothendieck classes of all melonic graphs are positive as polynomials in the class of the moduli space $mathcal M_{0,4}$. We also conjecture that the corresponding polynomials are log-concave, on the basis of hundreds of explicit computations.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"542 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77724772","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 symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends.
{"title":"Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations","authors":"Simone Floreani, F. Redig, F. Sau","doi":"10.1214/21-aihp1163","DOIUrl":"https://doi.org/10.1214/21-aihp1163","url":null,"abstract":"We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"43 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82517985","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：