arXiv subjects

Thomas Bothner

Publications and source records attributed to Thomas Bothner.

30 records · Page 2Linked to original sources

arXiv2016

From gap probabilities in random matrix theory to eigenvalue expansions

We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;dλ)\circlearrowleft$, acting on a single interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\{λ_i(J)\}_{i=0}^{\infty}$ of $K$ as $|J|\rightarrow\infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\det(I-γK)|_{L^2(J)}$ as $|J|\rightarrow\infty$ and $γ\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formulæ\, are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.

math-ph↗

arXiv2015

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential II

In this paper we continue our analysis \cite{BDIK} of the determinant $\det(I-γK_s),γ\in(0,1)$ where $K_s$ is the trace class operator acting in $L^2(-1,1)$ with kernel $K_s(λ,μ)=\frac{\sin s(λ-μ)}{π(λ-μ)}$. In \cite{BDIK} various key asymptotic results were stated and utilized, but without proof: Here we provide the proofs (see Theorem 1.2 and Proposition 1.3 below).

math-ph↗

arXiv2015

Hankel Determinant Approach to Generalized Vorob'ev-Yablonski Polynomials and their Roots

Generalized Vorob'ev-Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painlevé hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob'ev-Yablonski polynomials and provide formulæ\, for the boundary curves of the highly regular patterns observed numerically in \cite{CM}.

math-ph↗

arXiv2015

Transition asymptotics for the Painlevé II transcendent

We consider real-valued solutions $u=u(x|s),x\in\mathbb{R}$ of the second Painlevé equation $u_{xx}=xu+2u^3$ which are parametrized in terms of the monodromy data $s\equiv(s_1,s_2,s_3)\subset\mathbb{C}^3$ of the associated Flaschka-Newell system of rational differential equations. Our analysis describes the transition, as $x\rightarrow-\infty$, between the oscillatory power-like decay asymptotics for $|s_1|<1$ (Ablowitz-Segur) to the power-like growth behavior for $|s_1|=1$ (Hastings-McLeod) and from the latter to the singular oscillatory power-like growth for $|s_1|>1$ (Kapaev). It is shown that the transition asymptotics are of Boutroux type, i.e. they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant $\det(I-γK_{\textnormal{Ai}})|_{L^2(x,\infty)}$ in a double scaling limit $x\rightarrow-\infty,γ\uparrow 1$ as well as asymptotics for the spectrum of $K_{\textnormal{Ai}}$.

math-ph↗

arXiv2015

Universality conjecture and results for a model of several coupled positive-definite matrices

The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $(p+1)\times (p+1)$ matrix valued solution of a Riemann-Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields which include the Bessel$_ν$ universality for $1$-level and the Meijer-$G$ universality for $2$-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a $(p+1)\times (p+1)$ origin parametrix in terms of Meijer G-functions. The solution of the full Riemann-Hilbert problem is derived rigorously only for $p=3$ but the general framework of the proof can be extended to the Cauchy chain of arbitrary length $p$.

math-ph↗

arXiv2015

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I

We study the determinant $\det(I-γK_s), 0<γ<1$, of the integrable Fredholm operator $K_s$ acting on the interval $(-1,1)$ with kernel $K_s(λ, μ)= \frac{\sin s(λ- μ)}{π(λ-μ)}$. This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $β=2$, in the presence of an external potential $v=-\frac{1}{2}\ln(1-γ)$ supported on an interval of length $\frac{2s}π$. We evaluate, in particular, the double scaling limit of $\det(I-γK_s)$ as $s\rightarrow\infty$ and $γ\uparrow 1$, in the region $0\leqκ=\frac{v}{s}=-\frac{1}{2s}\ln(1-γ)\leq 1-δ$, for any fixed $0<δ<1$. This problem was first considered by Dyson in \cite{Dy1}.

math-ph↗

arXiv2014

Calculation of the constant factor in the six-vertex model

In the present paper we calculate explicitly the constant factor $C$ in the large $N$ asymptotics of the partition function $Z_N$ of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and ferroelectric phases. On the critical line the weights $a,b,c$ of the model are parameterized by a parameter $\al>1$, as $a=\frac{\al-1}{2}$, $b=\frac{\al+1}{2}$, $c=1$. The asymptotics of $Z_N$ on the critical line was obtained earlier in the paper \cite{BL2} of Bleher and Liechty: $Z_N=CF^{N^2}G^{\sqrt{N}}N^{1/4}\big(1+O(N^{-1/2})\big)$, where $F$ and $G$ are given by explicit expressions, but the constant factor $C>0$ was not known. To calculate the constant $C$, we find, by using the Riemann-Hilbert approach, an asymptotic behavior of $Z_N$ in the double scaling limit, as $N$ and $\al$ tend simultaneously to $\infty$ in such a way that $\frac{N}{\al}\to t\ge 0$. Then we apply the Toda equation for the tau-function to find a structural form for $C$, as a function of $\al$, and we combine the structural form of $C$ and the double scaling asymptotic behavior of $Z_N$ to calculate $C$.

math-ph↗

arXiv2014

Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity

In the present paper we derive a new Hankel determinant representation for the square of the Vorob'ev-Yablonski polynomial $\mathcal{Q}_n(x),x\in\mathbb{C}$. These polynomials are the major ingredients in the construction of rational solutions to the second Painlevé equation $u_{xx}=xu+2u^3+α$. As an application of the new identity, we study the zero distribution of $\mathcal{Q}_n(x)$ as $n\rightarrow\infty$ by asymptotically analyzing a certain collection of (pseudo) orthogonal polynomials connected to the aforementioned Hankel determinant. Our approach reproduces recently obtained results in the same context by Buckingham and Miller \cite{BM}, which used the Jimbo-Miwa Lax representation of PII equation and the asymptotical analysis thereof.

nlin.SI↗

arXiv2013

Asymptotics of a cubic sine kernel determinant

We study the one parameter family of Fredholm determinants $\det(I-γK_{\textnormal{csin}}),γ\in\mathbb{R}$ of an integrable Fredholm operator $K_{\textnormal{csin}}$ acting on the interval $(-s,s)$ whose kernel is a cubic generalization of the sine kernel which appears in random matrix theory. This Fredholm determinant appears in the description of the Fermi distribution of semiclassical non-equilibrium Fermi states in condensed matter physics as well as in random matrix theory. Using the Riemann-Hilbert method, we calculate the large $s$-asymptotics of $\det(I-γK_{\textnormal{csin}})$ for all values of the real parameter $γ$.

nlin.SI↗

arXiv2013

Tail decay for the distribution of the endpoint of a directed polymer

We obtain an asymptotic expansion for the tails of the random variable $\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where $\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \cite{Sch} that connects the density function of $\tcal$ to the Hastings-McLeod solution of the second Painlevé equation, we prove that as $t\rightarrow\infty$, $\mathbb{P}(|\tcal|>t)=Ce^{-4/3φ(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $φ(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly.

math-ph↗

arXiv2012

Asymptotics of a Fredholm determinant involving the second Painlevé transcendent

We study the determinant $\det(I-K_{\textnormal{PII}})$ of an integrable Fredholm operator $K_{\textnormal{PII}}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $Ψ$-function associated with the Hastings-McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-K_{\textnormal{PII}})$.

math-ph↗

arXiv2012

Exact solution of the six-vertex model with domain wall boundary conditions. Critical line between disordered and antiferroelectric phases

In the present article we obtain the large $N$ asymptotics of the partition function $Z_N$ of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and antiferroelectric phases. Using the weights $a=1-x,b=1+x,c=2,|x|<1$, we prove that, as $N\rightarrow\infty$, $Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$, where $F$ is given by an explicit expression in $x$ and the $x$-dependency in $C$ is determined. This result reproduces and improves the one given in the physics literature by Bogoliubov, Kitaev and Zvonarev. Furthermore, we prove that the free energy exhibits an infinite order phase transition between the disordered and antiferroelectric phases. Our proofs are based on the large $N$ asymptotics for the underlying orthogonal polynomials which involve a non-analytical weight function, the Deift-Zhou nonlinear steepest descent method to the corresponding Riemann-Hilbert problem, and the Toda equation for the tau-function.

math-ph↗