| 1. |
- Addario-Berry, Louigi, et al.
(författare)
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Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees
- 2013
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 41:2, s. 1072-1087
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Tidskriftsartikel (refereegranskat)abstract
- We study the height and width of a Galton-Watson tree with offspring distribution xi satisfying E xi = 1, 0 < Var xi < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.
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| 2. |
- Ahlberg, Daniel, et al.
(författare)
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To fixate or not to fixate in two-type annihilating branching random walks
- 2021
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 49:5, s. 2637-2667
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Tidskriftsartikel (refereegranskat)abstract
- We study a model of competition between two types evolving as branching random walks on Z(d). The two types are represented by red and blue balls, respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of Z(d) contain one ball each which are independently coloured red with probability p and blue otherwise. We address the question of fixation, referring to the sites and eventually settling for a given colour or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for p not equal 1/2 and that every site will change colour infinitely often almost surely for the balanced initial condition p = 1/2.
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| 3. |
- Alsmeyer, Gerold, et al.
(författare)
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The functional equation of the smoothing transform
- 2012
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 40:5, s. 2069-2105
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Tidskriftsartikel (refereegranskat)abstract
- Given a sequence T = (T-i)(i >= 1) of nonnegative random variables, a function f on the positive halfline can be transformed to E Pi(i >= 1) f (tT(i)). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on [0, infinity), the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, X =(d) Sigma(i >= 1) TiXi, where =(d) denotes equality of the corresponding laws, and X-1, X-2, ... is a sequence of i.i.d. copies of X independent of T. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation X =(d) inf{X-i/T-i:i >= 1, T-i > 0}. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.
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| 4. |
- 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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| 5. |
- Backhoff-Veraguas, Julio, et al.
(författare)
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Martingale Benamou–Brenier : A probabilistic perspective
- 2020
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Ingår i: Annals of Probability. - : The Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 48:5, s. 2258-2289
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Tidskriftsartikel (refereegranskat)abstract
- In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are corner-stones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport problem for given d-dimensional distributions mu, nu in convex order. The unique solution M* = (M-t*)(t is an element of[0,1]) of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the con ditions M-0*similar to mu, M-1*similar to nu. Similar to McCann's displacement-interpolation, M* provides a time-consistent interpolation between mu and nu. For particular choices of the initial and terminal law, M* recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer's theorem. This article is parallel to the work of Huesmann-Trevisan, who consider a related class of problems from a PDE-oriented perspective.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- Björklund, Michael
(författare)
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The asymptotic shape theorem for generalized first passage percolation
- 2010
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Ingår i: Annals of Probability. - 0091-1798 .- 2168-894X. ; 38:2, s. 632-660
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Björnberg, Jakob, 1983, et al.
(författare)
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STABLE SHREDDED SPHERES AND CAUSAL RANDOM MAPS WITH LARGE FACES
- 2022
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Ingår i: Annals of Probability. - 2168-894X .- 0091-1798. ; 50:5, s. 2056-2084
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a new familiy of random compact metric spaces Sα for α ∈ (1, 2), which we call stable shredded spheres. They are constructed from excursions of α-stable Lévy processes on [0, 1] possessing no negative jumps. Informally, viewing the graph of the Lévy excursion in the plane, each jump of the process is “cut open” and replaced by a circle, and then all points on the graph at equal height, which are not separated by a jump, are identified. We show that the shredded spheres arise as scaling limits of models of causal random planar maps with large faces introduced by Di Francesco and Guitter. We also establish that their Hausdorff dimension is almost surely equal to α. Point identification in the shredded spheres is intimately connected to the presence of decrease points in stable spectrally positive Lévy processes, as studied by Bertoin in the 1990s.
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| 10. |
- Blachere, S., et al.
(författare)
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A crossover for the bad configurations of random walk in random scenery
- 2011
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 39:5, s. 2018-2041
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case where the random walk has increments 0, +1 or -1 with probability epsilon, (1-epsilon)p and (1-epsilon)(1-p), respectively, with p is an element of [1/2, 1] and epsilon is an element of [0, 1), and where the scenery assigns the color black or white to the sites of Z with probability 1/2 each. We show that, remarkably, the set of bad configurations exhibits a crossover: for epsilon = 0 and p is an element of (1/2, 4/5) all configurations are bad, while for (p, epsilon) in an open neighborhood of (1, 0) all configurations are good. In addition, we show that for epsilon = 0 and p = 1/2 both bad and good configurations exist. We conjecture that for all epsilon is an element of [0, 1) the crossover value is unique and equals 4/5. Finally, we suggest an approach to handle the seemingly more difficult case where epsilon > 0 and p is an element of [1/2, 4/5), which will be pursued in future work.
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