scholarly journals Another probabilistic construction of Φ22n

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Yichao Huang
Keyword(s):  
Probabilistic Construction

scholarly journals On directed analogues of expander and hyperfinite graph sequences

2021 ◽  
pp. 1-14
Author(s):  
Endre Csóka ◽  
Łukasz Grabowski
Keyword(s):  
Directed Acyclic Graphs ◽  
Acyclic Graphs ◽  
Probabilistic Construction

Abstract We introduce and study analogues of expander and hyperfinite graph sequences in the context of directed acyclic graphs, which we call ‘extender’ and ‘hypershallow’ graph sequences, respectively. Our main result is a probabilistic construction of non-hypershallow graph sequences.

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

Discounted branching random walks

1985 ◽  
Vol 17 (01) ◽  
pp. 53-66
Author(s):  
K. B. Athreya
Keyword(s):  
Functional Equation ◽  
Random Walks ◽  
Unique Solution ◽  
Log P ◽  
Branching Random Walks ◽  
Probabilistic Construction ◽  
Image Position

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑∞ 0 pjsi a p.g.f. with p 0 = 0, < 1 < m = Σj p j < ∞ and 1 < ρ <∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x –θ ) for some θ > α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x –α ). We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x –α ) is relaxed.

Gaussian multiplicative chaos and Liouville quantum gravity

2018 ◽  
Author(s):  
Rémi Rhodes1 ◽  
Vincent Vargas2
Keyword(s):  
Quantum Gravity ◽  
Prior Knowledge ◽  
Scaling Limit ◽  
Planar Maps ◽  
Probabilistic Construction

The purpose of this chapter is to explain the probabilistic construction of Polyakov’s Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, this chapter contains a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. This chapter is rather short and requires no prior knowledge on the topic.

scholarly journals Randomized Construction of Complexes with Large Diameter

2020 ◽  
Author(s):  
Francisco Criado ◽  
Andrew Newman
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Large Diameter ◽  
Simplicial Complexes ◽  
Similar Improvement ◽  
Strongly Connected ◽  
Probabilistic Construction ◽  
Combinatorial Diameter

Abstract We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected $$(d-1)$$ ( d - 1 ) -dimensional simplicial complexes on n vertices, denoted $$H_s(n, d)$$ H s ( n , d ) . Using a probabilistic construction we give a new lower bound on $$H_s(n, d)$$ H s ( n , d ) that is within an $$O(d^2)$$ O ( d 2 ) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of $$e^{\varTheta (d)}$$ e Θ ( d ) of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

Sign in / Sign up

Export Citation Format

Share Document