Character expansion method for the first order asymptotics of a matrix integral
Alice Guionnet, Mylene Maida
备注：
Revised version : the organisation of the paper was changed quite deeply, namely we show a large deviation result for the empirical measures of Young tableaux in a bit more general context than dually weighted graphs models so that it can be used more easily to treat other models. In the concluding remarks, we provide another possible application of our result and a discussion about the links with maps enumeration. Some references are added
The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far. In this article, we follow an idea widely developped in the physics litterature, which is based on character expansion, to study more complex interaction. We more specifically consider a model defined in the spirit of the 'dually weighted graph model' studied by V. A. Kazakov, M. Staudacher and T. Wynter, but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices go to infinity and study the saddle points of the limit.
Expansions of Iterated Stratonovich Stochastic Integrals from the Taylor-Stratonovich Expansion, Based on Multiple Trigonometric Fourier Series. Comparison With the Milstein Expansion
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula.
Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
Asymptotic expansion of the distribution of a perturbation $Z_n$ of a Skorohod integral jointly with a reference variable $X_n$ is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable functions $f$ and also measurable functions $f$. In the latter case, the interpolation method connects the two non-degeneracies of variables for fini…
A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials
The article is devoted to the construction of effective procedures of the mean-square approximation of iterated Ito stochastic integrals of multiplicities 1 to 5 from the Taylor-Ito expansion based on multiple Fourier-Legendre series. The results of the article can be applied to the implementation of numerical methods with the orders of strong convergence 1.5, 2.0, and 2.5 for Ito stochastic differential equations with multidimensional non-commutative noise.
Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Exact Calculation of the Mean-Square Error in the Method of Expansion of Iterated Ito Stochastic integrals Based on Generalized Multiple Fourier Series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
The Proof of Convergence with Probability 1 in the Method of Expansion of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals of Multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich Expansions Using Legendre Polynomials
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
The article is devoted to optimization of the mean-square approximation procedures for iterated Ito stochastic integrals of multiplicities 1 to 5. The mentioned stochastic integrals are part of strong numerical methods with convergence orders 1.0, 1.5, 2.0, and 2.5 for Ito stochastic differential equations with multidimensional non-commutative noise based on the unified Taylor-Ito expansion and multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ $(k=1,\ldots,5).$ In this article we use multiple Fourier-Legendre series within the framework of the me…
The article is devoted to construction of effective procedures of the mean-square approximation for iterated Stratonovich stochastic integrals of multiplicities 1 to 5. We apply the method of generalized multiple Fourier series for approximation of iterated stochastic integrals. More precisely, we use multiple Fourier-Legendre series converging in the sense of norm in Hilbert space $L_2([t,T]^k),$ $k\in\mathbb{N}.$ Considered iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. That is why the results of the article can be applied to implementation of numer…
Expansions of Iterated Stratonovich Stochastic Integrals of Multiplicities 1 to 4. Combained Approach Based on Generalized Multiple and Iterated Fourier series
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
