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Finally cracked this egg.

Preprint here.

In my pursuit of an Ergodic Theorem for Random Walks on (probably finite) Quantum Groups, I have been looking at analogues of Irreducible and Periodic. I have, more or less, got a handle on irreducibility, but I am better at periodicity than aperiodicity.

The question of how to generalise these notions from the (finite) classical to noncommutative world has already been considered in a paper (whose title is the title of this post) of Fagnola and Pellicer. I can use their definition of periodic, and show that the definition of irreducible that I use is equivalent. This post is based largely on that paper.

## Introduction

Consider a random walk on a finite group driven by . The state of the random walk after steps is given by , defined inductively (on the algebra of functions level) by the associative

.

The convolution is also implemented by right multiplication by the stochastic operator:

,

where has entries, with respect to a basis . Furthermore, therefore

,

and so the stochastic operator describes the random walk just as well as the driving probabilty .

The random walk driven by is said to be *irreducible *if for all , there exists such that (if ) .

The *period *of the random walk is defined by:

.

The random walk is said to be *aperiodic *if the period of the random walk is one.

These statements have counterparts on the set level.

If is not irreducible, there exists a proper subset of , say , such that the set of functions supported on are -invariant. It turns out that is a proper subgroup of .

Moreover, when is irreducible, the period is the greatest common divisor of all the natural numbers such that there exists a partition of such that the subalgebras of functions supported in satisfy:

and (slight typo in the paper here).

In fact, in this case it is necessarily the case that is concentrated on a coset of a proper normal subgroup , say . Then .

Suppose that is supported on . We want to show that for . Recall that

.

This shows how the stochastic operator reduces the index .

A central component of Fagnola and Pellicer’s paper are results about how the decomposition of a stochastic operator:

,

specifically the maps can speak to the irreducibility and periodicity of the random walk given by . I am not convinced that I need these results (even though I show how they are applicable).

## Stochastic Operators and Operator Algebras

Let be a -algebra (so that is in general a virtual object). A -subalgebra is *hereditary *if whenever and , and , then .

It can be shown that if is a hereditary subalgebra of that there exists a projection such that:

.

All hereditary subalgebras are of this form.

In a recent preprint, Haonan Zhang shows that if (where is a Sekine Finite Quantum Group), then the convolution powers, , converges if

.

The algebra of functions is a multimatrix algebra:

.

As it happens, where , the counit on is given by , that is , dual to .

To help with intuition, making the incorrect assumption that is a classical group (so that is commutative — it’s not), because , the statement , implies that for a real coefficient ,

,

as for classical groups .

That is the condition is a quantum analogue of .

Consider a random walk on a classical (the algebra of functions on is commutative) *finite* group driven by a .

The following is a very non-algebra-of-functions-y proof that implies that the convolution powers of converge.

*Proof: *Let be the smallest subgroup of on which is supported:

.

We claim that the random walk on driven by is *ergordic* (see Theorem 1.3.2).

The driving probability is not supported on any proper subgroup of , by the definition of .

If is supported on a coset of proper normal subgroup , say , then because , this coset must be , but this also contradicts the definition of .

Therefore, converges to the uniform distribution on

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of .

So we have some questions:

- Is there a proof of the classical result (above) in the language of the algebra of functions on , that necessarily bypasses talk of points and of subgroups?
- And can this proof be adapted to the quantum case?
- Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

## Quantum Subgroups

Let be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication ) and a state on . We say that a quantum group with algebra of function (with comultiplication ) is a quantum subgroup of if there exists a surjective unital *-homomorphism such that:

.

## The Classical Case

In the classical case, where the algebras of functions on and are commutative,

There is a natural embedding, in the classical case, if is open (always true for finite) (thanks UwF) of ,

,

with for , and otherwise.

Furthermore, is has the property that

,

which resembles .

In the case where is a probability on a classical group , supported on a subgroup , it is very easy to see that convolutions remain supported on . Indeed, is the distribution of the random variable

,

where the i.i.d. . Clearly and so is supported on .

We can also prove this using the language of the commutative algebra of functions on , . The state being supported on implies that

.

Consider now two probabilities on but supported on , say . As they are supported on we have

and .

Consider

,

that is is also supported on and inductively .

## Some Questions

Back to quantum groups with non-commutative algebras of functions.

- Can we embed in with a map and do we have , giving the projection-like quality to ?
- Is a suitable definition for being supported on the subgroup .

If this is the case, the above proof carries through to the quantum case.

- If there is no such embedding, what is the appropriate definition of a being supported on a quantum subgroup ?
- If does not have the property of , in this or another definition, is it still true that being supported on implies that is too?

## Edit

UwF has recommended that I look at this paper to improve my understanding of the concepts involved.

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

**Abstract** *Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.*

## Distances between Probability Measures

Let be a finite quantum group and be the set of states on the -algebra .

The algebra has an invariant state , the dual space of .

Define a (bijective) map , by

,

for .

Then, where and , define the total variation distance between states by

.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative machinary shows that:

.

(supremum presentation)

In the classical case, using the test function , where , we have the probabilists’ preferred definition of total variation distance:

.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in , and therefore the classical total variation distance is equal to:

.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

## Quantum Projection Distance Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a factor of might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

Just back from a great workshop at Seoul National University, I am just going to use this piece to outline in a relaxed manner my key goals for my work on random walks on quantum groups for the near future.

In the very short term I want to try and get a much sharper lower bound for my random walk on the Sekine family of quantum groups. I believe the projection onto the ‘middle’ of the might provide something of use. On mature reflection, recognising that the application of the upper bound lemma is dominated by one set of terms in particular, it should be possible to use cruder but more elegant estimates to get the same upper bound except with lighter calculations (and also a smaller — see Section 5.7).

I also want to understand how sharp (or otherwise) the order convergence for the random walk on the dual of is — sounds awfully high. Furthermore it should be possible to get a better lower bound that what I have.

It should also be possible to redefine the quantum total variation distance as a supremum over projections subsets via . If I can show that for a positive linear functional that then using these ideas I can. More on this soon hopefully. No, this approach won’t work. (I have since completed this objective with some help: see here).

The next thing I might like to do is look at a random walk on the Sekine quantum groups with an -dependent driving probability and see if I can detect the cut-off phenomenon (Chapter 4). This will need good lower bounds for , some cut-off time.

Going back to the start, the classical problem began around 1904 with the question of Markov:

Which card shuffles mix up a deck of cards and cause it to ‘go random’?

For example, the perfect riffle shuffle does not mix up the cards at all while a riffle shuffle done by an amateur will.

In the context of random walks on classical groups this question is answered by the Ergodic Theorem 1.3.2: when the driving probability is not concentrated on a subgroup (irreducibility) nor the coset of a normal subgroup (aperiodicity).

Necessary and sufficient conditions on the driving probability for the random walk on a *quantum *group to converge to random are required. It is expected that the conditions may be more difficult than the classical case. However, it may be possible to use Diaconis-Van Daele theory to get some results in this direction. It should be possible to completely analyse some examples (such as the Kac-Paljutkin quantum group of order 8).

This will involve a study of subgroups of quantum groups as well as *normal *quantum subgroups.

It should be straightforward to extend the Upper Bound Lemma (Lemma 5.3.8) to the case of compact Kac algebras. Once that is done I will want to look at quantum generalisations of ‘natural’ random walks and shuffles.

I intend also to put the PhD thesis on the Arxiv. After this I have a number of options as regard to publishing what I have or maybe waiting a little while until I solve the above problems — this will all depend on how my further study progresses.

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

**Abstract** *A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. These ideas are generalised to the case of finite quantum groups.*

*Taken from Condition Expectation in Quantum Probabilty by Denes Petz.*

In quantum probability there are a number of fundamental questions that ask how faithfully can one quantise classical probability. Suppose that is a (classical) probability space and a sub--algebra. The *conditional expectation *of some integrable function (with respect to some -space) relative to is the orthogonal projection onto the closed subspace :

, .

Suppose now that is a quantum probability space and that is some C*-subalgebra of . Can we always define a conditional expectation with respect to ? The answer turns out to be not always, although this paper gives sufficient conditions for the existence of such a projection. Briefly, things work the other way around. Distinguished states give rise to quantum conditional expectations — and these conditional expectations define a subalgebra. We can’t necessarily start with a subalgebra and find the state which gives rise to it — Theorem 2 gives necessary conditions in which this approach does work.

*Taken from Real Analysis and Probability by R.M. Dudley.*

For a sequence of repeated, independent trials of an experiment, some probability distributions and variables converge as tends to infinity. In proving such limit theorems, it is useful to be able to construct a probability space on which a sequence of independent random variables is defined in a natural way; specifically, as coordinates for a countable Cartesian product.

The Cartesian product of finitely many -finite measure spaces gives a -finite measure space. For example, Cartesian products of Lesbesgue measure on the line give Lesbesgue measure on finite-dimensional Euclidean spaces. But suppose we take a measure space with two points each having measure , , and form a countable Cartesian product of copies of this space, so that the measure of any countable product of sets equals the product of their measures. Then we would get an uncountable space in which all singletons have measure , giving the measure usually called *counting measure*. An uncountable set with counting measure is not a -finite space, although in this example it was a countable product of finite measure spaces. By contrast, the the countable product of probability measures *will *again be a probability space. Here are some definitions.

For each let be a probability space. Let be the Cartesian product , that is, the set of all sequences with for all . Let be the natural projection of onto for each : for all . Let be the smallest -algebra of subsets of such that for all , is measurable from to . In other words, is the smallest -algebra containing all sets for all and all .

Let be the collection of all sets where for all and except for at most finitely many values of . Elements of will be called *rectangles*. Now recall the notion of semiring. has this property.

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