Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics, and spiked random matrices: Pinning and localization

Krajenbrink, Alexandre; Doussal, Pierre Le; O’Connell, Neil

doi:10.1103/physreve.103.042120

Cited by 16 publications

(20 citation statements)

References 128 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…with a t-independent term τ , the integration constant. But the same term was already computed in (166), hence consistency between (167) and (168) imposes the error term to be exponentially small, as claimed in (43). This completes our proof of Theorem 2.16.…”

Section: Riemannsupporting

confidence: 73%

See 1 more Smart Citation

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

Bothner¹

2022

Preprint

View full text Add to dashboard Cite

We characterize Fredholm determinants of a class of Hankel composition operators via matrixvalued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are not assumed to display the integrable structure known from the seminal work of Its, Izergin, Korepin and Slavnov [41]. Yet we are able to describe the corresponding Fredholm determinants through a naturally associated Riemann-Hilbert problem of Zakharov-Shabat type by solely exploiting the kernels' Hankel composition structures. We showcase the efficiency of this approach through a series of examples, we then compute several rank one perturbed determinants in terms of Riemann-Hilbert data and finally derive Akhiezer-Kac asymptotic theorems for suitable kernel classes.

show abstract

Section: Riemannsupporting

confidence: 73%

“…And these are just appearances of (1) and (2) random matrix theory, but they appear in other areas of mathematical physics, too, cf. [21,43].…”

Section: Introductionmentioning

confidence: 99%

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

Bothner¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…This implies that along any down right path in the quadrant Z 2 ⩾k , and up to a O(ε) error, the distribution of u, v ratios for the partition function Z ε−stat ℓ is the same as in the statement of proposition 3.5. Finally, equation (61) implies that the u, v ratios of Z ε−stat ℓ and Z stat ℓ have the same distribution as ε goes to zero, which implies the statement of the proposition. Now, fix some k ⩾ 1.…”

Section: Stationary Non-intersecting Log-gamma Polymersmentioning

confidence: 64%

A stationary model of non-intersecting directed polymers

Barraquand¹,

Doussal²

2023

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We consider the partition function $Z_{\ell}(\vec x,0\vert \vec y,t)$ of $\ell$ non-intersecting continuous directed polymers of length $t$ in dimension $1+1$, in a white noise environment, starting from positions $\vec x$ and terminating at positions $\vec y$. When $\ell=1$, it is well known that for fixed $x$, the field $\log Z_1(x,0\vert y,t)$ solves the Kardar-Parisi-Zhang equation and admits the Brownian motion as a stationary measure. In particular, as $t$ goes to infinity, $Z_1(x,0\vert y,t)/Z_1(x,0\vert 0,t) $ converges to the exponential of a Brownian motion $B(y)$. In this article, we show an analogue of this result for any $\ell$. We show that $Z_{\ell}(\vec x,0\vert \vec y,t)/Z_{\ell}(\vec x,0\vert \vec 0,t) $ converges as $t$ goes to infinity to an explicit functional $Z_{\ell}^{\rm stat}(\vec y)$ of $\ell$ independent Brownian motions. This functional $Z_{\ell}^{\rm stat}(\vec y)$ admits a simple description as the partition sum for $\ell$ non-intersecting semi-discrete polymers on $\ell$ lines. We discuss applications to the endpoints and midpoints distribution for long non-crossing polymers and derive explicit formula in the case of two polymers. To obtain these results, we show that the stationary measure of the O'Connell-Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric RSK correspondence found in \cite{corwin2014tropical}.

show abstract

“…This can be also interpreted using the fact that in the partition function Z stat 1 (n, m), if one sends parameter (n, m) to ∞ in the direction (Ψ 1 (̺ − λ), Ψ 1 (λ)), a simple computation, using the known explicit value of the free energy, shows that the optimal path will detach from the first row at a location such that the detaching point and the arrival point form an angle as defined by the vector (65). See similar arguments in [59,57,60].…”

Section: Interpretation In Terms Of Very Long Polymersmentioning

confidence: 94%

A stationary model of non-intersecting directed polymers

Barraquand¹,

Doussal²

2022

Preprint

View full text Add to dashboard Cite

We consider the partition function Z ℓ ( x, 0| y, t) of ℓ non-intersecting continuous directed polymers of length t in dimension 1+1, in a white noise environment, starting from positions x and terminating at positions y. When ℓ = 1, it is well known that for fixed x, the field log Z 1 (x, 0|y, t) solves the Kardar-Parisi-Zhang equation and admits the Brownian motion as a stationary measure. In particular, as t goes to infinity, Z 1 (x, 0|y, t)/Z 1 (x, 0|0, t) converges to the exponential of a Brownian motion B(y). In this article, we show an analogue of this result for any ℓ. We show that Z ℓ ( x, 0| y, t)/Z ℓ ( x, 0| 0, t) converges as t goes to infinity to an explicit functional Z stat ℓ ( y) of ℓ independent Brownian motion. This functional Z stat ℓ ( y) admits a simple description as the partition sum for ℓ non-intersecting semi-discrete polymers on ℓ lines. We discuss applications to the endpoints and midpoints distribution for long noncrossing polymers and derive explicit formula in the case of two polymers. To obtain these results, we show that the stationary measure of the O'Connell-Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric RSK correspondence found in [1]. Contents 1 Introduction and main results1 2 A stationary model for non-intersecting log-gamma polymers 12 KPZ limit 19Appendix: Maximum and argmax for the GUE(2) Dyson Brownian motion with drift 25

show abstract

Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics, and spiked random matrices: Pinning and localization

Cited by 16 publications

References 128 publications

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels

A stationary model of non-intersecting directed polymers

A stationary model of non-intersecting directed polymers

Contact Info

Product

Resources

About