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Publications (10 of 29) Show all publications
Johansson, K. (2019). The two-time distribution in geometric last-passage percolation. Probability Theory and Related Fields
Open this publication in new window or tab >>The two-time distribution in geometric last-passage percolation
2019 (English)In: Probability Theory and Related FieldsArticle in journal (Refereed) Published
Abstract [en]

We study the two-time distribution in directed last passage percolation with geometric weights in the first quadrant. We compute the scaling limit and show that it is given by a contour integral of a Fredholm determinant. 

Place, publisher, year, edition, pages
Springer New York LLC, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-252119 (URN)10.1007/s00440-019-00901-9 (DOI)2-s2.0-85062616626 (Scopus ID)
Note

QC 20190523

Available from: 2019-05-23 Created: 2019-05-23 Last updated: 2019-05-23Bibliographically approved
Beffara, V., Chhita, S. & Johansson, K. (2018). AIRY POINT PROCESS AT THE LIQUID-GAS BOUNDARY. Annals of Probability, 46(5), 2973-3013
Open this publication in new window or tab >>AIRY POINT PROCESS AT THE LIQUID-GAS BOUNDARY
2018 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 46, no 5, p. 2973-3013Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Institute of Mathematical Statistics, 2018
Keywords
Domino tilings, Airy kernel point process, two-periodic Aztec diamond
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-234572 (URN)10.1214/17-AOP1244 (DOI)000442612800013 ()2-s2.0-85052125071 (Scopus ID)
Note

QC 20180917

Available from: 2018-09-17 Created: 2018-09-17 Last updated: 2018-09-17Bibliographically approved
Adler, M., Johansson, K. & van Moerbeke, P. (2018). Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class. Mathematical physics, analysis and geometry, 21(1), Article ID 9.
Open this publication in new window or tab >>Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class
2018 (English)In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 21, no 1, article id 9Article in journal (Refereed) Published
Abstract [en]

This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties.

Place, publisher, year, edition, pages
Springer, 2018
Keywords
Lozenge tilings, Non-convex polygons, Kernels, Asymptotics
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-225184 (URN)10.1007/s11040-018-9265-5 (DOI)000427320600001 ()2-s2.0-85044122779 (Scopus ID)
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation, KAW 2010.0063
Note

QC 20180404

Available from: 2018-04-04 Created: 2018-04-04 Last updated: 2018-04-04Bibliographically approved
Adler, M., Johansson, K. & van Moerbeke, P. (2018). Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes. Communications in Mathematical Physics, 364(1), 287-342
Open this publication in new window or tab >>Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes
2018 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 364, no 1, p. 287-342Article in journal (Refereed) Published
Abstract [en]

This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model, we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.

Place, publisher, year, edition, pages
Springer, 2018
National Category
Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-237085 (URN)10.1007/s00220-018-3168-y (DOI)000446534900008 ()2-s2.0-85044153709 (Scopus ID)
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation, KAW 2010.0063
Note

QC 20181024

Available from: 2018-10-24 Created: 2018-10-24 Last updated: 2018-10-24Bibliographically approved
Chhita, S. & Johansson, K. (2016). Domino statistics of the two-periodic Aztec diamond. Advances in Mathematics, 294, 37-149
Open this publication in new window or tab >>Domino statistics of the two-periodic Aztec diamond
2016 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 294, p. 37-149Article in journal (Refereed) Published
Abstract [en]

Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary.

Place, publisher, year, edition, pages
Elsevier, 2016
Keywords
Domino tiling; Two-periodic Aztec diamond; Kasteleyn matrix; Asymptotics; Local statistics; Airy kernel
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-203634 (URN)10.1016/j.aim.2016.02.025 (DOI)000374207300002 ()2-s2.0-84960395062 (Scopus ID)
Note

QC 20170316

Available from: 2017-03-16 Created: 2017-03-16 Last updated: 2017-05-23Bibliographically approved
Chhita, S., Johansson, K. & Young, B. (2015). Asymptotic domino statistics in the Aztec diamond. The Annals of Applied Probability, 25(3), 1232-1278
Open this publication in new window or tab >>Asymptotic domino statistics in the Aztec diamond
2015 (English)In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 25, no 3, p. 1232-1278Article in journal (Refereed) Published
Abstract [en]

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

Keywords
Aztec diamond, Determinantal point process, Dimer covering, Domino tiling
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-166914 (URN)10.1214/14-AAP1021 (DOI)000353527000005 ()2-s2.0-84925438268 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation
Note

QC 20150529

Available from: 2015-05-29 Created: 2015-05-21 Last updated: 2017-12-04Bibliographically approved
Adler, M., Chhita, S., Johansson, K. & van Moerbeke, P. (2015). Tacnode GUE-minor processes and double Aztec diamonds. Probability theory and related fields, 162(1-2), 275-325
Open this publication in new window or tab >>Tacnode GUE-minor processes and double Aztec diamonds
2015 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 162, no 1-2, p. 275-325Article in journal (Refereed) Published
Abstract [en]

We study determinantal point processes arising in random domino tilings of a double Aztec diamond, a region consisting of two overlapping Aztec diamonds. At a turning point in a single Aztec diamond where the disordered region touches the boundary, the natural limiting process is the GUE-minor process. Increasing the size of a double Aztec diamond while keeping the overlap between the two Aztec diamonds finite, we obtain a new determinantal point process which we call the tacnode GUE-minor process. This process can be thought of as two colliding GUE-minor processes. As part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode GUE-minor process, we find the inverse Kasteleyn matrix for the dimer model version of the Double Aztec diamond.

Keywords
Airy process, Dimer, Extended kernels, Interlacing, Kasteleyn, Random Hermitian ensembles, Random tiling
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-170337 (URN)10.1007/s00440-014-0573-9 (DOI)000355182400008 ()2-s2.0-84929957197 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW 2010.0063Swedish Research Council
Note

QC 20150629

Available from: 2015-06-29 Created: 2015-06-29 Last updated: 2017-12-04Bibliographically approved
Adler, M., Johansson, K. & van Moerbeke, P. (2014). Double Aztec diamonds and the tacnode process. Advances in Mathematics, 252, 518-571
Open this publication in new window or tab >>Double Aztec diamonds and the tacnode process
2014 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 252, p. 518-571Article in journal (Refereed) Published
Abstract [en]

Discrete and continuous non-intersecting random processes have given rise to critical "infinite-dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the tacnode process. It is also-shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily.

Keywords
Domino tilings, Aztec diamonds, Dyson's Brownian motion, Airy and tacnode processes, Extended kernels
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-141956 (URN)10.1016/j.aim.2013.10.012 (DOI)000330153100019 ()2-s2.0-84889675905 (Scopus ID)
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation, KAW 2010.0063
Note

QC 20140228

Available from: 2014-02-28 Created: 2014-02-27 Last updated: 2017-12-05Bibliographically approved
Johansson, K. (2013). Non-colliding Brownian Motions and the Extended Tacnode Process. Communications in Mathematical Physics, 319(1), 231-267
Open this publication in new window or tab >>Non-colliding Brownian Motions and the Extended Tacnode Process
2013 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, no 1, p. 231-267Article in journal (Refereed) Published
Abstract [en]

We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.

Keywords
Random-Matrix Theory, Determinantal Processes, Pearcey Process, Orthogonal Polynomials, Strong Asymptotics, Wigner Matrices, Universality, Ensembles
National Category
Physical Sciences Mathematics
Identifiers
urn:nbn:se:kth:diva-121109 (URN)10.1007/s00220-012-1600-2 (DOI)000316490100006 ()
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation, KAW2010.0063
Note

QC 20130422

Available from: 2013-04-22 Created: 2013-04-19 Last updated: 2017-12-06Bibliographically approved
Johansson, K. (2012). Universality for certain Hermitian Wigner matrices under weak moment conditions. Annales de l'I.H.P. Probabilites et statistiques, 48(1), 47-79
Open this publication in new window or tab >>Universality for certain Hermitian Wigner matrices under weak moment conditions
2012 (English)In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 48, no 1, p. 47-79Article in journal (Refereed) Published
Abstract [en]

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

Keywords
Wigner matrix, Gaussian divisible, Optimal moment condition, Universality, Tracy-Widom distribution
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-91262 (URN)10.1214/11-AIHP429 (DOI)000300338800002 ()2-s2.0-84856276225 (Scopus ID)
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation, KAW2010.0063
Note
QC 20120312Available from: 2012-03-12 Created: 2012-03-12 Last updated: 2017-12-07Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-2943-7006

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