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  • 1. Ameur, Yacin
    et al.
    Hedenmalm, Håkan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Makarov, Nikolai
    Random normal matrices and ward identities2015Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 43, nr 3, s. 1157-1201Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

  • 2.
    Bao, Zhigang
    et al.
    Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Clear Water Bay, Hong Kong, Peoples R China..
    Erdos, Laszlo
    IST Austria, Campus 1, A-3400 Klosterneuburg, Austria..
    Schnelli, Kevin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE2019Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 47, nr 3, s. 1270-1334Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Sigma be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189-1217] asserts that the empirical eigenvalue distribution of the matrix X : = U Sigma V* converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in C. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N-1/2+epsilon and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).

  • 3.
    Beffara, Vincent
    et al.
    Univ Grenoble Alpes, CNRS, Inst Fourier, F-38000 Grenoble, France..
    Chhita, Sunil
    Univ Durham, Dept Math Sci, Durham DH1 3LE, England..
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Royal Inst Technol KTH, Dept Math, Linstedtsvagen 25, SE-10044 Stockholm, Sweden..
    AIRY POINT PROCESS AT THE LIQUID-GAS BOUNDARY2018Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 46, nr 5, s. 2973-3013Artikel i tidskrift (Refereegranskat)
    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.

  • 4.
    Björklund, Michael
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    The asymptotic shape theorem for generalized first passage percolation2010Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, nr 2, s. 632-660Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We generalize the asymptotic shape theorem in first passage percolation on Z(d) to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for Z(d)-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon-Nikodym property.

  • 5. Buckdahn, Rainer
    et al.
    Djehiche, Boualem
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    Li, Juan
    Peng, Shige
    MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS: A LIMIT APPROACH2009Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 37, nr 4, s. 1524-1565Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

  • 6.
    Duits, Maurice
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    ON GLOBAL FLUCTUATIONS FOR NON-COLLIDING PROCESSES2018Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 46, nr 3, s. 1279-1350Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.

  • 7.
    Erhardsson, Torkel
    KTH, Tidigare Institutioner, Matematik.
    Strong memoryless times and rare events in Markov renewal point processes2004Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 32, nr 3B, s. 2446-2462Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let W be the number of points in (0, t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable (, we construct a "strong memoryless time" zeta such that zeta - t is exponentially distributed conditional on {zeta less than or equal to t, zeta > t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon-Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.

  • 8. Hachem, Walid
    et al.
    Hardy, Adrien
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Najim, Jamal
    Large complex correlated wishart matrices: Fluctuations and asymptotic independence at the edges2016Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 44, nr 3, s. 2264-2348Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale N-2/3. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin ( hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale N-2.

  • 9. Hult, Henrik
    et al.
    Lindskog, Filip
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    Extremal behavior of stochastic integrals driven by regularly varying Levy processes2007Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 35, nr 1, s. 309-339Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the extremal behavior of a stochastic integral driven by a multivariate Levy process that is regularly varying with index alpha > 0. For predictable integrands with a finite (alpha + delta)-moment, for some delta > 0, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Levy process and we determine its limit measure associated with regular variation on the space of cadlag functions.

  • 10.
    Hägg, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Local Gaussian fluctuations in the Airy and Discrete PNG processes2008Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 36, nr 3, s. 1059-1092Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove that the Airy process, A(t), locally fluctuates like a Brownian motion. In the same spirit we also show that, in a certain scaling limit, the so-called discrete polynuclear growth process (PNG) behaves like a Brownian motion.

  • 11. Janson, Svante
    et al.
    Stefánsson, Sigurdur Örn
    KTH, Centra, Nordic Institute for Theoretical Physics NORDITA.
    Scaling limits of random planar maps with a unique large face2015Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 43, nr 3, s. 1045-1081Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n(-1/2) is described by a Brownian excursion. The planar maps, with the graph metric resealed by n(-1/2), are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

  • 12.
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The arctic circle boundary and the Airy process2005Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 33, nr 1, s. 1-30Artikel i tidskrift (Refereegranskat)
  • 13.
    Johansson, Kurt
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Lambert, Gaultier
    Univ Zurich, Inst Math, Winterthurerstr 190, CH-8057 Zurich, Switzerland..
    GAUSSIAN AND NON-GAUSSIAN FLUCTUATIONS FOR MESOSCOPIC LINEAR STATISTICS IN DETERMINANTAL PROCESSES2018Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 46, nr 3, s. 1201-1278Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study mesoscopic linear statistics for a class of determinantal point processes which interpolate between Poisson and random matrix statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the Gaussian Unitary Ensemble (GUE) eigenvalue process. An example of such a system comes from considering the distribution of noncolliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the correlation kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we obtain a central limit theorem (CLT) of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we observe a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sinekernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.

  • 14.
    Karlsson, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ledrappier, Francois
    On laws of large numbers for random walks2006Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 34, nr 5, s. 1693-1706Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also generalizes Oseledec's multiplicative ergodic theorem. In addition, we show that epsilon-shadows of any ballistic random walk with finite moment on any group eventually intersect. Some related results concerning Coxeter groups and mapping class groups are recorded in the last section.

  • 15. Nyström, Kaj
    et al.
    Önskog, Thomas
    The Skorohod oblique reflection problem in time-dependent domains2010Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, nr 6, s. 2170-2223Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka [61] and further investigated by P.-L. Lions and A.-S. Sznitman [42] in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of cadlag solutions (x, lambda) to the Skorohod problem, with oblique reflection, for (D, Gamma, w) assuming, in particular, that D is a time-dependent domain (Theorem 1.2). In addition, we prove that if w is continuous, then x is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.

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