Edge fluctuations of limit shapes

Johansson, Kurt

doi:10.4310/cdm.2016.v2016.n1.a2

Cited by 27 publications

(41 citation statements)

References 64 publications

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“…Those include the Pearcey kernel [78] for quartic singularities (Airy is cubic), or the tacnode kernel [79] (which includes, roughly speaking, quadratic band touching). We refer to [80] for a review of these free fermionic universality classes.…”

Section: A1 Tuning the Dispersion Relationmentioning

confidence: 99%

Free fermions at the edge of interacting systems

Stéphan

2019

SciPost Phys.

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We study the edge behavior of inhomogeneous one-dimensional quantum systems, such as Lieb-Liniger models in traps or spin chains in spatially varying fields. For free systems these fall into several universality classes, the most generic one being governed by the Tracy-Widom distribution. We investigate in this paper the effect of interactions. Using semiclassical arguments, we show that since the density vanishes to leading order, the strong interactions in the bulk are renormalized to zero at the edge, which simply explains the survival of Tracy-Widom scaling in general. For integrable systems, it is possible to push this argument further, and determine exactly the remaining length scale which controls the variance of the edge distribution. This analytical prediction is checked numerically, with excellent agreement. We also study numerically the edge scaling at fronts generated by quantum quenches, which provide new universality classes awaiting theoretical explanation.

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Section: A1 Tuning the Dispersion Relationmentioning

confidence: 99%

Free fermions at the edge of interacting systems

Stéphan

2019

SciPost Phys.

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“…More explicitly, suppose that E = {e i } r i=1 is a collection of distinct edges with e i = (b i , w i ), where b i and w i denote black and white vertices. 24,23]). The dimers form a determinantal point process on the edges of the Aztec diamond graph with correlation kernel L meaning that…”

Section: Inverse Kasteleyn Matrix and The Particle Processmentioning

confidence: 99%

Airy point process at the liquid-gas boundary

Beffara¹,

Chhita²,

Johansson³

2018

Ann. Probab.

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Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

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“…Ensembles of non-intersecting random lines, both in the discrete and continuous setups, as well as their scaling limits as the linear size of the system grows, play a significant role in the probabilistic analysis of various problems in random matrices, interacting particle systems and effective interface models; see e.g. [10,17,22,8,2,1,18,24,7,9] and references therein.…”

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confidence: 99%