Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.
We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $\mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
Given a sequence $(ξ_n)$ of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper ``Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process'' {\it Acta Math.} (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series $f(z) = \sum_{n=1}^\infty ξ_n z^{n-1}$ in the complex unit disc $\mathbb{D}$ constitute a determinantal point process. The study of the zeros of the general random series $f(z)$ where the restriction of independence is relaxed upon the random variables $(ξ_n)$ is an important open problem. Th…
We study mesoscopic linear statistics for a class of determinantal point processes which interpolates between Poisson and Gaussian Unitary Ensemble statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the GUE eigenvalue process. An example of such a system comes from considering the distribution of non-colliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the GUE kernel, related to the size of the cylinder o…
We consider a collection of Euclidean random balls in ${\Bbb R}^d$ generated by a determinantal point process inducing interaction into the balls. We study this model at a macros\-copic level obtained by a zooming-out and three different regimes --Gaussian, Poissonian and stable-- are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.
Consider an inhomogeneous multi-species TASEP with drift to the left, and define a height function which equals the maximum species number to the left of a lattice site. For each fixed time, the multi-point distributions of these height functions have a determinantal structure. In the homogeneous case and for certain initial conditions, the fluctuations of the height function converge to Gaussian random variables in the large-time limit. The proof utilizes a coupling between the multi-species TASEP and a coalescing random walk, and previously known results for coalescing random walks.
National University of Singapore（新加坡国立大学）；Massachusetts Institute of Technology（麻省理工学院）
Determinantal point processes (a.k.a. DPPs) have recently become popular tools for modeling the phenomenon of negative dependence, or repulsion, in data. However, our understanding of an analogue of a classical parametric statistical theory is rather limited for this class of models. In this work, we investigate a parametric family of Gaussian DPPs with a clearly interpretable effect of parametric modulation on the observed points. We show that parameter modulation impacts the observed points by introducing directionality in their repulsion structure, and the principal directions correspond to…
We consider two-dimensional determinantal processes which are rotation-invariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry-Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal process,…
In certain point processes, the configuration of points outside a bounded domain determines, with probability 1, certain statistical features of the points within the domain. This notion, called rigidity, was introduced in a work of Ghosh and Peres. In this paper, rigidity and the related notion of tolerance are examined systematically and point processes with rigidity of various degrees are introduced. Natural classes of point processes such as determinantal point processes, zero sets of Gaussian entire functions and perturbed lattices are examined from the point of view of rigidity, and gene…
School of Mathematics, University of Bristol（布里斯托大学数学学院）
Determinantal points processes are a promising but relatively under-developed tool in machine learning and statistical modelling, being the canonical statistical example of distributions with repulsion. While their mathematical formulation is elegant and appealing, their practical use, such as simply sampling from them, is far from straightforward.Recent work has shown how a particular type of determinantal point process defined on the compact multidimensional space $[-1, 1]^d$ can be practically sampled and further shown how such samples can be used to improve Monte Carlo integration.This wor…
The first main result of this note, Theorem 1.2, establishes the determinantal identities (7) and (8) for the expectation, under a determinantal point process governed by an integrable projection kernel, of scaling limits of characteristic polynomials sampled at several points. The determinantal identities (7) and (8) can be seen as the scaling limit of the identity of Fyodorov and Strahov for the averages of ratios of products of the values of the characteristic polynomial of a Gaussian unitary matrix. Borodin, Olshanski and Strahov derived the determinantal identity of Fyodorov and Strahov f…
We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.
We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of of a log-gas confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic statistics. Namely, at small scales, the point process still behaves according to random matrix theory, while, at large scales, it needs to be renormalized because the variance o…
We consider the noncolliding Brownian motion (BM) with $N$ particles starting from the eigenvalue distribution of Gaussian unitary ensemble (GUE) of $N \times N$ Hermitian random matrices with variance $σ^2$. We prove that this process is equivalent with the time shift $t \to t+σ^2$ of the noncolliding BM starting from the configuration in which all $N$ particles are put at the origin. In order to demonstrate nontriviality of such equivalence for determinantal processes, we show that, even from its special consequence, determinantal expressions are derived for the ensemble averages of products…
The Ginibre point process is given by the eigenvalue distribution of a non-hermitian complex Gaussian matrix in the infinite matrix-size limit. This is a determinantal point process (DPP) on the complex plane ${\mathbb{C}}$ in the sense that all correlation functions are given by determinants specified by an integral kernel called the correlation kernel. Shirai introduced the one-parameter ($m \in {\mathbb{N}}_0$) extensions of the Ginibre DPP and called them the Ginibre-type point processes. In the present paper we consider a generalization of the Ginibre and the Ginibre-type point processes …
The bead process introduced by Boutillier is a countable interlacing of the determinantal sine-kernel point processes. We construct the bead process for general sine beta processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite beta corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian unitary and orthogonal ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circul…
We consider the multi-time correlation and covariance structure of a random surface growth with a wall introduced in arXiv:0904.2607. It is shown that the correlation functions associated with the model along space-like paths have determinantal structure, which yields the convergence of height fluctuations to that of a Gaussian free field. We also construct a continuous-time non-commutative random walk on $U(\mathfrak{so}_{N+1})$, which matches the random surface growth when restricting to the Gelfand-Tsetlin subalgebra of $U(\mathfrak{so}_{N+1})$. As an application, we prove the convergence o…
Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
我们研究了三角形网格上多边形均匀菱形拼接所对应的随机台阶表面的大尺度高度波动。对于一类多边形（允许任意大的边数），我们表明，这些波动是渐近高斯自由（无质量）场。这补充了在Kenyon [Comm.Math.Phys.281（2008）675-709]中获得的关于没有极限形状的冻结小平面的区域的平铺的类似结果。在我们的渐近分析中，我们使用显式双围道积分公式用于先前在Petrov [Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes（2012）Preprint]中获得的模型的行列式相关核。
Central limit theorem for random partitions under the Plancherel measure
在这项工作中，我们得到的中心极限定理波动的杨图周围的极限形状在大部分的“频谱”的分区的一个大整数n（下的Plancherel措施）。更具体地说，我们表明，在适当的规范化（增长的平方根的log n），相应的随机过程收敛，在有限维分布的意义上，高斯过程的独立值。证明大量使用相关函数的行列式结构，并基于应用的Costin-Lebowitz-Soshnikov中心极限定理。在光谱边缘，波动渐近表示的艾里系综的最大成员，特别是在上边缘的极限分布似乎是离散的（没有任何规范化）。这些结果承认一个优雅的对称下旋转45度的杨氏图，归一化不再依赖于频谱点的位置重新制定。我们还讨论了联系我们的中心极限定理与较早的结果由S. V. Kerov的收敛到一个广义高斯过程。
The zeros of the random Laurent series $1/μ- \sum_{j=1}^\infty c_j/z^j$, where each $c_j$ is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement $z \mapsto 1/z$, we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the cor…
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers,whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be in…
We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of…
We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used. In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on t…
