From gap probabilities in random matrix theory to eigenvalue expansions

Bothner, Thomas

doi:10.1088/1751-8113/49/7/075204

Cited by 16 publications

(9 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since we were able to derive an exact determinant formula for it, the return probability is perhaps the simplest observable for which a rigorous long time analysis can be performed. Indeed, similar determinants have been studied using differential equations [78,79], operator theoretic [67,80], or Riemann-Hilbert techniques [55,56,59,81]. While such methods fall outside the scope of the present paper, a proof of the nowhere continuous behavior, together with the exponential decay away from roots of unity is left as a pressing issue.…”

Section: Resultsmentioning

confidence: 99%

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

Stéphan¹

2017

J. Stat. Mech.

View full text Add to dashboard Cite

We study the return probability and its imaginary (τ) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy |∆| < 1. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to τ 2 , which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the region |∆| < 1 the decay for large times t is nowhere continuous as a function of anisotropy: it is either gaussian at root of unity or exponential otherwise. As an aside, we also determine that the front moves as x f (t) = t √ 1 − ∆ 2 , by analytic continuation of known arctic curves in the six vertex model. Exactly at |∆| = 1, we find the return probability decays as e −ζ(3/2) √ t/π t 1/2 O(1). It is argued that this result provides an upper bound on spin transport. In particular, it suggests that transport should be diffusive at the isotropic point for this quench. arXiv:1707.06625v3 [cond-mat.stat-mech]

show abstract

Section: Resultsmentioning

confidence: 99%

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

Stéphan¹

2017

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…As τ approaches 0 and 2 3 √ 2 the oscillations die out, though due to different mechanisms in each case. [Bot17,Bot16] managed to translate his asymptotic result for u AS into a corresponding result for F only in the (c) region 9 . For region (a), [Bot17] demonstrated a simplified form of u AS (x; γ(x)) for τ ∈ 0, (−x) −δ for any fixed δ > 0.…”

Section: Asymptotics Of Ablowitz-segur Solution Of Painlevé IImentioning

confidence: 99%

“…based on the lack of oscillations in uAS for such τ , and[Bot16] provided an extension to the full region (c) (and slightly beyond).Corwin & Ghosal/Lower tail of the KPZ equation…”

mentioning

confidence: 95%

Lower tail of the KPZ equation

Corwin,

Ghosal

2018

Preprint

View full text Add to dashboard Cite

We provide the first tight bounds on the lower tail probability of the one point distribution of the KPZ equation with narrow wedge initial data. Our bounds hold for all sufficiently large times T and demonstrates a crossover between super-exponential decay with exponent 5/2 (and leading pre-factor 4 15π T 1/3 ) for tail depth greater than T 2/3 , and exponent 3 (with leading pre-factor 1 12 ) for tail depth less than T 2/3 .

show abstract

“…Note that, the equation (1.27) reduces to a special case of the Painlevé V equation via the transformation y(τ ) = (q(τ 2 ) − 1)/(q(τ 2 ) + 1). Furthermore, the following large s asymptotics of det(I − K Bes ) is conjectured in [38], and later rigorously proved in [20] (see also [6]):…”

Section: Gap Probability At the Hard Edgementioning

confidence: 94%

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

Dai¹,

Xu²,

Zhang³

2018

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We study the Fredholm determinant of an integrable operator acting on the interval (0, s) whose kernel is constructed out of the Ψ-function associated with a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order k in a certain scaling regime. Using the Riemann-Hilbert method, we obtain the large s asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to a coupled Painlevé III system.

show abstract

From gap probabilities in random matrix theory to eigenvalue expansions

Cited by 16 publications

References 45 publications

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

Lower tail of the KPZ equation

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

Contact Info

Product

Resources

About