Fine Asymptotics Research Articles

Smooth Solutions of the tt* Equation: A Numerical Aided Case Study

An important special class of the tt* equations are the tt*-Toda equations. Guest et al. have given comprehensive studies on the tt*-Toda equations in a series of papers. The fine asymptotics for a large class of solutions of a special tt*-Toda equation, the case 4a in their classification, have been obtained in the paper [Comm. Math. Phys. 374 (2020), 923-973] in the series. Most of these formulas are obtained with elaborate reasoning and the calculations involved are lengthy. There are concerns about these formulas if they have not been verified by other methods. The first part of this paper is devoted to the numerical verification of these fine asymptotics. In fact, the numerical studies can do more and should do more. A natural question is whether we can find more such beautiful formulas in the tt* equation via numerical study. The second part of this paper is devoted to the numerical study of the fine asymptotics of the solutions in an enlarged class defined from the Stoke data side. All the fine asymptotics of the solutions in the enlarged class are found by the numerical study. The success of the numerical study is largely due to the truncation structures of the tt* equation.

Hearing the shape of ancient noncollapsed flows in R4$\mathbb {R}^{4}$

AbstractWe consider ancient noncollapsed mean curvature flows in whose tangent flow at is a bubble‐sheet. We carry out a fine spectral analysis for the bubble‐sheet function u that measures the deviation of the renormalized flow from the round cylinder and prove that for we have the fine asymptotics , where is a symmetric 2 × 2‐matrix whose eigenvalues are quantized to be either 0 or . This naturally breaks up the classification problem for general ancient noncollapsed flows in into three cases depending on the rank of Q. In the case , generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or 2d‐bowl. In the case , under the additional assumption that the flow either splits off a line or is self‐similarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be 2d‐oval or belongs to the one‐parameter family of 3d oval‐bowls constructed by Hoffman‐Ilmanen‐Martin‐White, respectively. Finally, in the case we show that the flow is compact and SO(2)‐symmetric and for has the same sharp asymptotics as the O(2) × O(2)‐symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.

Large Deviation Estimates of Selberg’s Central Limit Theorem and Applications

Abstract For $V\sim \alpha \log \log T$ with $0<\alpha <2$, we prove $$\begin{align*} & \frac{1}{T}\textrm{meas}\{t\in [T,2T]: \log|\zeta(1/2+ \textrm{i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^{2}/\log\log T}. \end{align*}$$This improves prior results of Soundararajan and of Harper on the large deviations of Selberg’s Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł, and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $(\log T)^{\theta }$, $0<\theta <3$, that is expected to be sharp for $\theta> 0$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł, and one of the authors to prove fine asymptotics for the maximum on intervals of length $1$.

Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded and their distribution function satisfies a mild regularity condition near zero. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight distribution near the largest possible value and in certain cases we manage to find the corrections up to random order. Additionally, we describe the tail distribution of the maximum degree, the distribution of the number of vertices attaining the maximum degree, and establish asymptotic normality of the number of vertices with near-maximum degree. Our analysis extends the results proved for random recursive trees (where the weights are constant) to the case of random weights. The main technical result shows that the degrees of several uniformly chosen vertices are asymptotically independent with explicit error corrections.

Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees

We prove an invariance principle for linearly edge reinforced random walks on γ-stable critical Galton-Watson trees, where γ∈(1,2] and where the edge joining x to its parent has rescaled initial weight d(O,x)α for some α≤1. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.

Fisher-KPP equation with small data and the extremal process of branching Brownian motion

We consider the limiting extremal process X of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale Z, the rescaled “density” of particles, which are at distance n+x from a position close to the tip of X, converges in probability to a multiple of the exponential ex as n→+∞. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a 1-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in [9,10] and further developed in the present paper. The proofs of the limit theorems for X rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of “size” 0<ε≪1, up to terms of the order [(log⁡ε−1)]−1−γ, with some γ>0.

Fine asymptotics for models with Gamma type moments

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the [Formula: see text]-dimensional volume of the simplex with [Formula: see text] points in [Formula: see text] distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-[Formula: see text] convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].

Corrections to “Fine Asymptotics for Universal One-to-One Compression of Parametric Sources” [Apr 19 2442-2458

In The paper above [1] , published in the April 2019 issue of the IEEE Transactions on Information Theory, the following change is noted. The affiliations of the authors were listed incorrectly due to a production error. The affiliations should be listed as follows: The authors were with the School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287 USA, while performing their research for [1] .

Local theorems for arithmetic compound renewal processes when Cramer's condition holds

We continue the study of the compound renewal processes (c.r.p.), where the moment Cramer's condition holds (see [1]-[10], where the study of c.r.p. was started). In the paper arithmetic c.r.p. Z(n) are studied. In such processes random vector {\xi} = ({\tau},{\zeta}) has the arithmetic distribution, where {\tau} > 0 defines the distance between jumps, {\zeta} defines the values of jumps. For this processes the fine asymptotics in the local limit theorem for probabilities P(Z(n) = x) has been obtained in Cramer's deviation region of x $\in$ Z. In [6]-[10] the similar problem has been solved for non-lattice c.r.p., when the vector {\xi} = ({\tau},{\zeta}) has the non-lattice distribution.

Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation

We give here an explicit formula for the following critical case of the growth-fragmentation equation ∂/∂tu(t,x)+∂/∂x(gxu(t,x))+bu(t,x)=bα2u(t,αx), u(0,x)=u0(x) for some constants g &gt; 0, b &gt; 0 and α &gt;1 - the case α = 2 being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in [8], and use them to clarify how periodicity may appear asymptotically.

Radial fast diffusion on the hyperbolic space

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.

Fine asymptotics of Christoffel functions for general measures

Let u be a measure on the unit circle satisfying Szegő's condition. In 1991, A. Mate calculated precisely the first-order asymptotic behavior of the sequence of Christoffel functions associated with u when the point of evaluation lies on the circle, resolving a long-standing open problem. We extend his results to measures supported on smooth curves in the plane. In the process, we derive new asymptotic estimates for the Cesaro means of the corresponding 1-Faber polynomials and investigate some applications to orthogonal polynomials, linear ill-posed problems and the mean ergodic theorem.

Convergence of U-Statistics for Interacting Particle Systems

The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, de99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework.

Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

We investigate local and global properties of positive solutions to the fast diffusion equation in the good exponent range , corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space , we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time.