arXiv subjects

Renjie Feng

Publications and source records attributed to Renjie Feng.

31 records · Page 2Linked to original sources

arXiv2017

Critical radius and supremum of random spherical harmonics

We first consider {\it deterministic} immersions of the $d$-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level $n$. The main result of the article is the, a priori unexpected, fact that there is a uniform lower bound to the critical radius of the immersions as $n\to\infty$. This fact has immediate implications for {\it random} spherical harmonics with fixed $L^2$-norm. In particular, it leads to an exact and explicit formulae for the tail probability of their (large deviation) suprema by the tube formula, and also relates this to the expected Euler characteristic of their upper level sets.

math.PR↗

arXiv2016

Correlations between zeros and critical points of random analytic functions

We study the two-point correlation $K^m_n(z,w)$ between zeros and critical points of Gaussian random holomorphic sections $s_n$ over Kähler manifolds. The critical points are points $\nabla_{h^n} s_n=0$ where $\nabla_{h^n}$ is the smooth Chern connection with respect to the Hermitian metric $h^n$ on line bundle $L^n$. The main result is that the rescaling limit of $K^m_n(z_0+\frac u{\sqrt n}, z_0+\frac v{\sqrt n})$ for any $z_0\in M$ is universal as $n$ tends to infinity. In fact, the universal rescaling limit is the two-point correlation between zeros and critical points of Gaussian analytic functions for the Bargmann-Fock space of level $1$. Furthermore, there is a 'repulsion' between zeros and critical points for the short range; and a 'neutrality' for the long range.

math.PR↗

arXiv2015

Conditional expectations of random holomorphic fields on Riemann surfaces

We study two conditional expectations: the expected density of critical points of Gaussian random holomorphic sections of powers of a positive holomorphic line bundle over Riemann surfaces given that the random sections vanish at a point and the expected density of zeros given that the random sections has a fixed critical point. The main result is that these two rescaling conditional expectations have universal limits as the power of the line bundle tends to infinity. We will see that the short distance behavior between critical points and the conditioning zero is neutral while there is a repulsion between zeros and the conditioning critical point. But the long distance behaviors are the same.

math.CV↗

arXiv2015

Critical values of fixed Morse index of random analytic functions on Riemann surfaces

This note is an addendum to 'Critical values of random analytic functions on complex manifolds, Indiana Univ. Math. J. 63 No. 3 (2014), 651-686.' by R.Feng and S. Zelditch (arXiv:1212.4762). In this note, we give the formula of the limiting distribution of the critical values at critical points of random analytic functions of fixed Morse index and we compute explicitly such limitings for local maxima and saddle values in the case of Riemann surfaces; we also prove that the second order term in the full expansion of the distribution of the critical values is topologically invariant on Riemann surfaces.

math.PR↗

arXiv2015

Critical values of Gaussian SU(2) random polynomials

In this note, we will get the estimate of the expected distribution of critical values of Gaussian SU(2) random polynomials as the degree large enough. The result is a direct application of the Kac-Rice formula. The critical values will accumulate at infinity, we further study the rate of this convergence and its rescaling limit as $n\to\infty$.

math.PR↗

arXiv2013

Median and mean of the Supremum of $L^2$ normalized random holmorphic fields

We prove that the expected value and median of the supremum of $L^2$ normalized random holomorphic fields of degree $n$ on $m$-dimensional Kähler manifolds are asymptotically of order $\sqrt{m\log n}$. This improves the prior result of Shiffman-Zelditch (arXiv:math/0303335) that the upper bound of the media is of order $\sqrt{\log n}$ The estimates are based on the entropy methods of Dudley and Sudakov combined with a precise analysis of the relevant pseudo-metric and its covering numbers, which can be precisely evaluated using off-diagonal asymptotics of Bergman kernels. Recent work of the authors on the value distribution of these fields are also used to get precise constants.

math.PR↗

arXiv2012

Critical values of random analytic functions on complex manifolds

We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a positive Hermitian holomorphic line bundle $(L, h). In the special case M = CP^m and L = O(1), and h is the Fubini-Study metric, the random polynomials are the SU(m + 1) polynomials. By a critical value we mean the norm ||s_n||_h of s_n at a non-zero critical point of the norm. The metric and Kahler form endow the polynomials with a Hilbert space structure and we consider the associated Gaussian random polynomials and the spherical ensemble where ||s_n|| = 1 is chosen from Haar measure. Our main result is that the expected limit distribution of critical values as n tends to infinity in the spherical ensemble is universal (i.e. is independent of the choice of h), and we give an explicit formula for it. The limit distribution is the same as for suitably normalized Gaussian measures.

math.PR↗

arXiv2012

The global existence and convergence of the Calabi flow on $\mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$

In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on $X$ when the dimension is 2, partially confirming Chen's conjecture. Moreover, we show that the Calabi flow exponentially converges to the flat Kähler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldson's conjecture.

math.DG↗

arXiv2011

Random Riesz energies on compact Kähler manifolds

This article determines the asymptotics of the expected Riesz s-energy of the zero set of a Gaussian random systems of polynomials of degree N as the degree N tends to infinity in all dimensions and codimensions. The asymptotics are proved more generally for sections of any positive line bundle over any compact Kaehler manifold. In comparison with the results on energies of zero sets in one complex dimension due to Qi Zhong (arXiv:0705.2000) (see also [arXiv:0705.2000]), the zero sets have higher energies than randomly chosen points in dimensions > 2 due to clumping of zeros.

math.PR↗

arXiv2011

Periodic solutions of Abreu's equation

We solve Abreu's equation with periodic right hand side, in any dimension. This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety.

math.AP↗

arXiv2010

Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_θ)$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized Szasz analytic function $S_{h^N}(f)(x)$. About $S_{h^N}(f)(x)$, we prove that it admits complete asymptotics and there exists a universal scaling limit. % We also consider some dilation operator composed with $S_{h^N}(f)(x)$ and we give an estimate about this composition. As an example, we will further compute $S_{h^N}(f)(x)$ for the Bergman metric on the unit ball.

math.DG↗

arXiv2010

Bergman metrics and geodesics in the space of Kähler metrics on principally polarized Abelian varieties

It's well-known in \kahler geometry that the infinite dimensional symmetric space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized \kahler manifold is well approximated by finite dimensional submanifolds $\bcal_k \subset \hcal$ of Bergman metrics of height $k$. Then it's natural to ask whether geodesics in $\hcal$ can be approximated by Bergman geodesics in $\bcal_k$. For any polarized \kahler manifold, the approximation is in the $C^0$ topology. While Song-Zelditch proved the $C^2$ convergence for the torus-invariant metrics over toric varieties. In this article, we show that some $C^{\infty}$ approximation exists as well as a complete asymptotic expansion for principally polarized Abelian varieties. We also get a $C^\infty$ complete asymptotic expansion for harmonic maps into $\bcal_k$ which generalizes the work of Rubinstein-Zelditch on toric varieties.

math.DG↗

arXiv2010

Large deviations for zeros of $P(ϕ)_2$ random polynomials

We extend results of Zeitouni-Zelditch on large deviations principles for zeros of Gaussian random polynomials $s$ in one complex variable to certain non-Gaussian ensembles that we call $P(ϕ)_2$ random polynomials. The probability measures are of the form $e^{- S(f)} df$ where the actions $S(f)$ are finite dimensional analgoues of those of $P(ϕ)_2$ quantum field theory. The speed and rate function are the same as in the associated Gaussian case. As a corollary, we prove that the expected distribution of zeros in the $P(ϕ)_2$ ensembles tends to the same equilibrium measure as in the Gaussian case.

math.PR↗