There is no stationary \(p\)-cyclically monotone Poisson matching in 2d. (English) Zbl 07922274
Summary: We show that for \(p > 1\) there is no \(p\)-cyclically monotone stationary matching of two independent Poisson processes in dimension \(d = 2\). The proof combines the \(p\)-harmonic approximation result from [15, Theorem 1.1] with local asymptotics for the two-dimensional matching problem. Moreover, we prove a.s. local upper bounds of the correct order in the case \(p > 1\), which, to the best of our knowledge, are not readily available in the current literature.
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
| 49Q22 | Optimal transportation |
Keywords:
allocation; cyclical monotonicity; invariance; matching; optimal transport; \(p\)-minimal; stationarityReferences:
| [1] | M. Ajtai, J. Komlós, and G. Tusnády. On optimal matchings. Combinatorica, 4:259-264, 1984. MathSciNet: MR0779885 · Zbl 0562.60012 |
| [2] | F. Barthe and C. Bordenave. Combinatorial optimization over two random point sets. In Séminaire de probabilités XLV, pages 483-535. Cham: Springer, 2013. MathSciNet: MR3185927 · Zbl 1401.90180 |
| [3] | S. Boucheron, G. Lugosi, and P. Massart. Concentration inequalities. Oxford University Press, Oxford, 2013. MathSciNet: MR3185193 · Zbl 1279.60005 |
| [4] | S. Chatterjee, R. Peled, Y. Peres, and D. Romik. Gravitational allocation to Poisson points. Ann. Math., pages 617-671, 2010. MathSciNet: MR2680428 · Zbl 1206.60013 |
| [5] | M. Goldman, M. Huesmann, and F. Otto. Quantitative linearization results for the Monge-Ampère equation. Com. Pure and Appl. Math, 2021. MathSciNet: MR4373162 |
| [6] | M. Goldman and D. Trevisan. Convergence of asymptotic costs for random Euclidean matching problems. Probab. Math. Phys., 2020. MathSciNet: MR4408015 |
| [7] | C. E. Gutiérrez and A. Montanari. \( L^{\operatorname{\infty}} \)-estimates in optimal transport for non quadratic costs. Calc. Var. Partial Differ. Equ., 61(5):163, Jun 2022. MathSciNet: MR4444176 · Zbl 07547959 |
| [8] | C. Hoffman, A. E. Holroyd, and Y. Peres. A stable marriage of Poisson and Lebesgue. Ann. Probab., 34(4):1241-1272, 2006. MathSciNet: MR2257646 · Zbl 1111.60008 |
| [9] | A. E. Holroyd. Geometric properties of Poisson matchings. Probab. Theory Relat. Fields, 150(3):511-527, 2011. MathSciNet: MR2824865 · Zbl 1225.60082 |
| [10] | A. E. Holroyd, S. Janson, and J. Wästlund. Minimal matchings of point processes. Probab. Theory Relat. Fields, 184(1):571-611, Oct 2022. MathSciNet: MR4498517 · Zbl 1500.60006 |
| [11] | A. E. Holroyd, R. Pemantle, Y. Peres, and O. Schramm. Poisson matching. Ann. Inst. Henri Poincaré, Probab. Stat., 45(1):266-287, 2009. MathSciNet: MR2500239 · Zbl 1175.60012 |
| [12] | M. Huesmann. Optimal transport between random measures. Ann. Inst. Henri Poincaré, Probab. Stat., 52(1):196 - 232, 2016. MathSciNet: MR3449301 · Zbl 1332.60029 |
| [13] | M. Huesmann, F. Mattesini, and F. Otto. There is no stationary cyclically monotone Poisson matching in 2d. Probab. Theory Relat. Fields, August 2023. MathSciNet: MR4664582 |
| [14] | M. Huesmann and K.-T. Sturm. Optimal transport from Lebesgue to Poisson. Ann. Probab., 41(4):2426-2478, 2013. MathSciNet: MR3112922 · Zbl 1279.60024 |
| [15] | L. Koch. Geometric regularisation for optimal transport with strongly p-convex cost Calc. Var. Partial Differ. Equ. , 63(4): 87, 2024 MathSciNet: MR4728221 · Zbl 1540.49055 |
| [16] | R. Markó and Á. Timár. A Poisson allocation of optimal tail. Ann. Probab., 44(2):1285-1307, 2016. MathSciNet: MR3474472 · Zbl 1338.60027 |
| [17] | F. Santambrogio. Optimal transport for applied mathematicians, volume 87 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling. MathSciNet: MR3409718 · Zbl 1401.49002 |
| [18] | G. Schwarz. Hodge Decomposition - A Method for Solving Boundary Value Problems. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2006. MathSciNet: MR1367287 |
| [19] | E. M. Stein. Singular Integrals and Differentiability Properties of Functions (PMS-30). Princeton University Press, 1970. MathSciNet: MR0290095 · Zbl 0207.13501 |
| [20] | C. Villani. Optimal transport: old and new, volume 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 2009. MathSciNet: MR2459454 · Zbl 1156.53003 |
| [21] | L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Relat. Fields, 118(3):427-438, 2000. MathSciNet: MR1800540 · Zbl 0970.60093 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
