| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- Budzinski, Thomas, et al.
(författare)
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LOCAL LIMITS OF BIPARTITE MAPS WITH PRESCRIBED FACE DEGREES IN HIGH GENUS
- 2022
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 50:3, s. 1059-1126
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Tidskriftsartikel (refereegranskat)abstract
- We study the local limits of uniform high genus bipartite maps with prescribed face degrees. We prove the convergence toward a family of infinite maps of the plane, the q-IBPMs, which exhibit both a spatial Markov property and a hyperbolic behaviour. Therefore, we observe a similar local behaviour for a wide class of models of random high genus maps which can be seen as a result of universality. Our results cover all the regimes where the expected degree of the root face remains finite in the limit. This follows a work by the same authors on high genus triangulations.
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| 5. |
- Janson, Svante, 1955-, et al.
(författare)
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Unicellular maps vs. hyperbolic surfaces in large genus : Simple closed curves
- 2023
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 51:3, s. 899-929
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Tidskriftsartikel (refereegranskat)abstract
- We study uniformly random maps with a single face, genus g, and size n, as n,g → ∞ with g = o(n), in continuation of several previous works on the geometric properties of “high genus maps.” We calculate the number of short simple cycles, and we show convergence of their lengths (after a well-chosen rescaling of the graph distance) to a Poisson process, which happens to be exactly the same as the limit law obtained by Mirzakhani and Petri (Comment. Math. Helv. 94 (2019) 869–889) when they studied simple closed geodesics on random hyperbolic surfaces under the Weil–Petersson measure as g → ∞.This leads us to conjecture that these two models are somehow “the same” in the limit, which would allow to translate problems on hyperbolic surfaces in terms of random trees, thanks to a powerful bijection of Chapuy, Féray and Fusy (J. Combin. Theory Ser. A 2013 (120) 2064–2092).
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| 6. |
- Johansson, Kurt, 1960-, et al.
(författare)
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Multivariate Normal Approximation For Traces Of Random Unitary Matrices
- 2021
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 49:6, s. 2961-3010
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Tidskriftsartikel (refereegranskat)abstract
- In this article we obtain a superexponential rate of convergence in total variation between the traces of the first m powers of a n x n random unitary matrices and a 2m-dimensional Gaussian random variable. This generalizes previous results in the scalar case to the multivariate setting, and we also give the precise dependence on the dimensions m and n in the estimates with explicit constants. We are especially interested in the regime where m grows with n and our main result basically states that if m <
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| 7. |
- Johansson, Kurt, 1960-, et al.
(författare)
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On inhomogeneous polynuclear growth
- 2022
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 50:2, s. 559-590
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Tidskriftsartikel (refereegranskat)abstract
- This article studies the inhomogeneous geometric polynuclear growth model; the distribution of which is related to Schur functions. We explain a method to derive its distribution functions in both space-like and time-like directions, focusing on the two-time distribution. Asymptotics of the two-time distribution in the KPZ-scaling limit is then considered, extending to two times several single-time distributions in the KPZ universality class.
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| 8. |
- Sjöstrand, Jonas
(författare)
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MONOTONE SUBSEQUENCES IN LOCALLY UNIFORM RANDOM PERMUTATIONS
- 2023
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 51:4, s. 1502-1547
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Tidskriftsartikel (refereegranskat)abstract
- A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the j th point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under the correct scaling, for any r ≥ 0, the largest union of (Formula Presenmted)decreasing subsequences approaches a limit surface as n tends to infinity, and the limit surface is a solution to a specific variational problem. As a corollary, we prove the existence of a limit shape for the Young diagram associated to the random permutation under the Robinson– Schensted correspondence. In the special case where ρ is the uniform distribution on the diamond |x| + |y| < 1, we conjecture that the limit shape is triangular, and assuming the conjecture is true, we find an explicit formula for the limit surfaces of a uniformly random permutation and recover the famous limit shape of Vershik, Kerov and Logan, Shepp.
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| 9. |
- Thévenin, Paul
(författare)
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A GEOMETRIC REPRESENTATION OF FRAGMENTATION PROCESSES ON STABLE TREES
- 2021
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Ingår i: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 49:5, s. 2416-2476
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Tidskriftsartikel (refereegranskat)abstract
- We provide a new geometric representation of a family of fragmentation processes by nested laminations which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation, obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous-Pitman fragmentation of the Brownian continuum random tree and minimal factorizations of the n-cycle, that is, factorizations of the permutation (12...n) into a product of (n-1) transpositions, proving this way a conjecture of Feray and Kortchemski. We discuss various properties of these new lamination-valued processes, and we notably show that they can be coded by explicit Levy processes.
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