| 1. |
- Ameur, Yacin, et al.
(författare)
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Random normal matrices and ward identities
- 2015
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 43:3, s. 1157-1201
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Bao, Zhigang, et al.
(författare)
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LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE
- 2019
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 47:3, s. 1270-1334
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Tidskriftsartikel (refereegranskat)abstract
- 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).
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| 3. |
- Beffara, Vincent, et al.
(författare)
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AIRY POINT PROCESS AT THE LIQUID-GAS BOUNDARY
- 2018
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 46:5, s. 2973-3013
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Tidskriftsartikel (refereegranskat)abstract
- 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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| 4. |
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| 5. |
- Cotar, Codina, et al.
(författare)
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Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs
- 2017
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Ingår i: Annals of Probability. - 0091-1798. ; 45:4, s. 2655-2706
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. 19 (2009) 1972-2007] for finite graphs with edge reinforcement. We apply our new method both to edge- and to vertex-reinforced random walks with super-linear reinforcement on arbitrary infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs. For edge-reinforced random walks, we complete the results of Limic and Tarrès [Ann. Probab. 35 (2007) 1783-1806] and we settle a conjecture of Sellke (1994) by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times. For vertex-reinforced random walks, we extend results previously obtained on Z by Volkov [Ann. Probab. 29 (2001) 66-91] and by Basdevant, Schapira and Singh [Ann. Probab. 42 (2014) 527-558], and on complete graphs by Benaim, Raimond and Schapira [ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 767-782]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times.
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| 6. |
- Duits, Maurice
(författare)
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ON GLOBAL FLUCTUATIONS FOR NON-COLLIDING PROCESSES
- 2018
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Ingår i: Annals of Probability. - : INST MATHEMATICAL STATISTICS. - 0091-1798 .- 2168-894X. ; 46:3, s. 1279-1350
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Hachem, Walid, et al.
(författare)
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Large complex correlated wishart matrices : Fluctuations and asymptotic independence at the edges
- 2016
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 44:3, s. 2264-2348
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- Janson, Svante, et al.
(författare)
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Scaling limits of random planar maps with a unique large face
- 2015
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 43:3, s. 1045-1081
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Johansson, Kurt, et al.
(författare)
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GAUSSIAN AND NON-GAUSSIAN FLUCTUATIONS FOR MESOSCOPIC LINEAR STATISTICS IN DETERMINANTAL PROCESSES
- 2018
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Ingår i: Annals of Probability. - : INST MATHEMATICAL STATISTICS. - 0091-1798 .- 2168-894X. ; 46:3, s. 1201-1278
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Lubetzky, Eyal, et al.
(författare)
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Strong Noise Sensitivity and Random Graphs
- 2015
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 43:6, s. 3239-3278
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Tidskriftsartikel (refereegranskat)abstract
- The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm (1999), it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability. Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise given a specific witness in the original input. Our main context is the Erdos-Renyi random graph, where already the property of containing a given graph is sufficiently rich to separate these notions. In particular, our analysis implies (strong) noise sensitivity in settings where the BKS criterion involving the first Fourier level does not apply, e.g., when the critical value goes to 0 polynomially fast in the number of variables.
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