Invariance principles for random walks in cones

Duraj, Jetlir; Wachtel, Vitali

doi:10.48550/arxiv.1508.07966

Cited by 5 publications

(7 citation statements)

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“…Conditional limit theorems and conditional invariance principles for random walks in different cones have been studied in [6] and [8]. All the results in these papers are proved in the case when the non-rescaled walk starts at a fixed point.…”

Section: Appendix B Invariance Principles For Random Walks In Weyl Ch...mentioning

confidence: 99%

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Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Ioffe

Velenik

Wachtel

2017

Probab. Theory Relat. Fields

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Abstract. We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of 2 + 1-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari-Spohn diffusions associated with limiting Sturm-Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.

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Section: Appendix B Invariance Principles For Random Walks In Weyl Ch...mentioning

confidence: 99%

“…The organization of these sections is described in Subsection 3.8. Many of our technical estimates rely on strong approximation techniques and on a refinement of recent results on random walks in Weyl chambers and on cones [6,8].…”

Section: Introductionmentioning

confidence: 99%

Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Ioffe

Velenik

Wachtel

2017

Probab. Theory Relat. Fields

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“…Since the geometric assumption (ii) has been used in [7] in the construction of V (x) only, Theorem 1 allows us to state limit theorems for random walks in cones proven in [7] and in [10] for all cones satisfying (i).…”

Section: Introduction and The Main Resultmentioning

confidence: 99%

Alternative constructions of a harmonic function for a random walk in a cone

Denisov¹,

Wachtel²

2019

Electron. J. Probab.

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For a random walk killed at leaving a cone we suggest two new constructions of a positive harmonic function. These constructions allow one to remove a quite strong extendability assumption, which has been imposed in our previous paper (Denisov and Wachtel, 2015, Random walks in cones). As a consequence, all the limit results from that paper remain true for cones which are either convex or star-like and C 2 .

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“…This Lemma will prove useful as cone-points events imply the events in the left-hand side of (15). First, by (8) and the definition of Y , we have…”

Section: 3mentioning

confidence: 98%

“…Proof of Theorem 5.3. The above changes in the arguments from [10] allow one to repeat the proof of [15,Theorem 6], which gives the convergence of a properly centered and rescaled walk S n towards the two-dimensional Brownian bridge conditioned to stay in the upper half-plane. This convergence is uniform in the range of u, v as formulated in Theorem 5.3.…”

Section: Proof Ofmentioning

confidence: 99%