Random walks on graphs and approximation of L²-Invariants
In this work, we interpret right multiplication operators Rw: l2(G) → l2(G) , w∈ ℂ[G] as random walk operators on certain labelled graphs we employ that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. I. Grigorchuk and A. Żuk to these graphs gives a n...
Saved in:
Main Authors: | , |
---|---|
Other Authors: | |
Format: | Article |
Language: | English |
Published: |
2022
|
Subjects: | |
Online Access: | https://hdl.handle.net/10356/160048 |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Institution: | Nanyang Technological University |
Language: | English |
id |
sg-ntu-dr.10356-160048 |
---|---|
record_format |
dspace |
spelling |
sg-ntu-dr.10356-1600482022-07-12T02:55:22Z Random walks on graphs and approximation of L²-Invariants Kricker, Andrew Wong, Zenas School of Physical and Mathematical Sciences Science::Mathematics Random Walks Spectral Density Function In this work, we interpret right multiplication operators Rw: l2(G) → l2(G) , w∈ ℂ[G] as random walk operators on certain labelled graphs we employ that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. I. Grigorchuk and A. Żuk to these graphs gives a new interpretation and proof of a special case of W. Lück’s famous Theorem on the Approximation of L2-Betti numbers for countable residually finite groups by means of exhausting towers of finite-index subgroups. In particular, using this interpretation, the theorem follows naturally from standard theorems in probability theory concerning the weak convergence of probability measures that are characterized by their moments. This paper is mainly a direct adaptation of the ideas of Grigorchuk, Zuk̇ and Lück to this setting. We aim to explain how these ideas are related and give a short exposition of them. Nanyang Technological University This research was supported by the Nanyang Technological University Academic Research Fund Tier 1 Grant RG 32/17. 2022-07-12T02:55:22Z 2022-07-12T02:55:22Z 2021 Journal Article Kricker, A. & Wong, Z. (2021). Random walks on graphs and approximation of L²-Invariants. Acta Mathematica Vietnamica, 46(2), 309-319. https://dx.doi.org/10.1007/s40306-021-00425-2 0251-4184 https://hdl.handle.net/10356/160048 10.1007/s40306-021-00425-2 2-s2.0-85105446414 2 46 309 319 en RG 32/17 Acta Mathematica Vietnamica © 2021 Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. |
institution |
Nanyang Technological University |
building |
NTU Library |
continent |
Asia |
country |
Singapore Singapore |
content_provider |
NTU Library |
collection |
DR-NTU |
language |
English |
topic |
Science::Mathematics Random Walks Spectral Density Function |
spellingShingle |
Science::Mathematics Random Walks Spectral Density Function Kricker, Andrew Wong, Zenas Random walks on graphs and approximation of L²-Invariants |
description |
In this work, we interpret right multiplication operators Rw: l2(G) → l2(G) , w∈ ℂ[G] as random walk operators on certain labelled graphs we employ that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. I. Grigorchuk and A. Żuk to these graphs gives a new interpretation and proof of a special case of W. Lück’s famous Theorem on the Approximation of L2-Betti numbers for countable residually finite groups by means of exhausting towers of finite-index subgroups. In particular, using this interpretation, the theorem follows naturally from standard theorems in probability theory concerning the weak convergence of probability measures that are characterized by their moments. This paper is mainly a direct adaptation of the ideas of Grigorchuk, Zuk̇ and Lück to this setting. We aim to explain how these ideas are related and give a short exposition of them. |
author2 |
School of Physical and Mathematical Sciences |
author_facet |
School of Physical and Mathematical Sciences Kricker, Andrew Wong, Zenas |
format |
Article |
author |
Kricker, Andrew Wong, Zenas |
author_sort |
Kricker, Andrew |
title |
Random walks on graphs and approximation of L²-Invariants |
title_short |
Random walks on graphs and approximation of L²-Invariants |
title_full |
Random walks on graphs and approximation of L²-Invariants |
title_fullStr |
Random walks on graphs and approximation of L²-Invariants |
title_full_unstemmed |
Random walks on graphs and approximation of L²-Invariants |
title_sort |
random walks on graphs and approximation of l²-invariants |
publishDate |
2022 |
url |
https://hdl.handle.net/10356/160048 |
_version_ |
1738844889637978112 |