In this article, we develop a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the limits of spherical integrals obtained in [46,47]. As examples, we obtain
1. a large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary Haar distributed matrices $U$;
2. a large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at measures which are described by the free product with amalgamation;
3. a large deviation principle for the Kostka number $K_{\boldsymbolλ_N \boldsymbolη_N}$, for two sequences of partitions $\boldsymbolλ_N, \boldsymbolη_N$ with at most $N$ rows;
4. a large deviation upper bound for the Littlewood-Richardson coefficients $c_{\boldsymbolλ_N \boldsymbol η_N}^{\boldsymbol κ_N}$, for three sequences of partitions $\boldsymbolλ_N, \boldsymbol η_N, \boldsymbol κ_N$ with at most $N$ rows, and their complementary lower bounds at nice measures.
In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as $m,n$ go to infinity while $m/n$ converges.
We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thäle (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.
We study the spherical model of a ferromagnet in $d$-dimensional cubes $Ω_n$ of volume $|Ω_n|=n^d$ and investigate large deviations of the magnetization of various domains $D_k\subset Ω_n$. We focus our attention on the low-temperature regime, $T<T_c$, and consider domains $D_k$ of three types: $(d-1)$-dimensional layers of width $k$, $(d-2)$-dimensional rods, and Kadanoff blocks. In the case of layers the large-deviation probabilities decay exponentially with $n^{d-2}$, and we obtain an explicit expression for the corresponding rate function. When the layer width $k\ll n$, the large-deviation…
We consider the asymptotics of $k$-dimensional spherical integrals when $k = o(N)$. We prove that the $o(N)$-dimensional spherical integrals are approximately the products of $1$-dimensional spherical integrals. Our formulas extend the results for $k$-dimensional spherical integrals proved by Guionnet and Maïda in [29] and Husson and Guionnet in [34] which are only valid for $k$ finite and independent of $N$. These approximations will be used to prove a large deviation principle for the joint $2k(N)$ extreme eigenvalues for sharp sub-Gaussian Wigner matrices and for additive deformations of GO…
We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in the asymptotic expansion. Using asymptotic expression of the spherical integral, we derive the asymptotic freeness of Wigner matrices with (deterministic) Hermitian matrices.
For suitable families of locally infinitely divisible Markov processes $\{ξ^{ε}_t\}_{0\leq t\leq T}$ with frequent small jumps depending on a small parameter $ε>0,$ precise asymptotics for large deviations of integral forms $\mathbb{E}^ε[\exp\{ε^{-1}F(ξ^ε)\}]$ are proved for smooth functionals $F.$ The main ingredient of the proof in this paper is a recent result regarding the asymptotic expansions of the expectations $\mathbb{E}^ε[G(ξ^ε)\}]$ for smooth $G.$ Several connections between these large deviation asymptotics and partial integro-differential equations are included as well.
For a fractional Brownian motion $B^H$ with Hurst parameter $H\in]{1/4},{1/2}[\cup]{1/2},1[$, multiple indefinite integrals on a simplex are constructed and the regularity of their sample paths are studied. Then, it is proved that the family of probability laws of the processes obtained by replacing $B^H$ by $ε^{1/2} B^H$ satisfies a large deviation principle in Hölder norm. The definition of the multiple integrals relies upon a representation of the fractional Brownian motion in terms of a stochastic integral with respect to a standard Brownian motion. For the large deviation principle, the a…
The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles…
In this note, we study the asymptotics of a spherical integral that is a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. This counterpart can also be expressed in terms the Heckman-Opdam hypergeometric function. When the argument of this spherical integral is of finite support and of order $N$, these asymptotics involve a modified version of the $S$-transform of the limit measure of the matrix argument and its largest eigenvalue. To prove our main result, we are leveraging a technique of successive conditionning. In particular we prove in a "mathematically …
We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.
R-transforms for non-Hermitian block random matrices: a spherical integral approach
Institut Louis Bachelier（路易·巴舍利耶研究所）；Econophysics Lab, Institut Louis Bachelier（经济物理学实验室，路易·巴舍利耶研究所）；LadHyX, UMR CNRS 7646, École polytechnique, Institut Polytechnique de Paris（拉迪克斯实验室，法国国家科学研究中心联合研究单位7646，巴黎综合理工学院，巴黎理工学院）
In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$ tend to infinity, of the logarithm (divided by $n$) of the expectation of $\exp[\sqrt{nm}θX_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $θ$ is a real number, $M_n$ is a certain $n\times m$ deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal or unitary matr…
We show that for a suitable class of functions of finitely-many variables, the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.
In this article, we consider flat and curved Riemannian symmetric spaces in the complex case and we study their basic integral kernels, in potential and spherical analysis: heat, Newton, Poisson kernels and spherical functions, i.e. the kernel of the spherical Fourier transform. We introduce and exploit a simple new method of construction of these $W$-invariant kernels by alternating sum formulas. We then use the alternating sum representation of these kernels to obtain their asymptotic behavior. We apply our results to the Dyson Brownian Motion on $R^d$.
R-transforms for non-Hermitian matrices: a spherical integral approach
Econophysics Lab, Institut Louis Bachelier（经济物理实验室，路易·巴舍利耶研究所）；LadHyX, UMR CNRS 7646, Ecole Polytechnique, Institut Polytechnique de Paris（拉迪希克斯实验室，法国国家科学研究中心联合研究单位第7646号，巴黎综合理工学院，巴黎理工学院）；Capital Fund Management（资本基金管理公司）
A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are consid…
In this paper, we consider the addition of two matrices in generic position, namely A + U BU * , where U is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices A and B, the law of the largest eigenvalue satisfies a large deviation principle, in the scale N, with an explicit rate function involving the limit of spherical integrals. We cover in particular all the cases when A and B have no outliers.
We establish a comprehensive sample path large deviation principle (LDP) for log-processes associated with multivariate time-inhomogeneous stochastic volatility models. Examples of models for which the new LDP holds include Gaussian models, non-Gaussian fractional models, mixed models, models with reflection, and models in which the volatility process is a solution to a Volterra type stochastic integral equation. The LDP for log-processes is used to obtain large deviation style asymptotic formulas for the distribution function of the first exit time of a log-process from an open set and for th…
We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at $\infty$ with exponent $θ$. In applications this parameter can provide information on fractal properties of processes that are subordinate to $x(\cdot)$. For this reason the estimation of $θ$ is an important theoretical problem. Here, we consider the process $x(t)$ whose derivative is fractional Brownian motion with self-similarity parameter $0<H<1$. For this case…
