Background

I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by \nu\in M_p(G) is ergodic if the convolution powers (\nu^{\star k})_{k\geq 0} converge to the Haar state \int_G.

The classical theorem for finite groups:

Ergodic Theorem for Random Walks on Finite Groups

A random walk on a finite group G driven by a probability \nu\in M_p(G) is ergodic if and only if \nu is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.

Not concentrated on a proper subgroup gives irreducibility. A random walk is irreducible if for all g\in G, there exists k\in\mathbb{N} such that \nu^{\star k}(\{g\})>0.

Not concentrated on the coset of a proper normal subgroup gives aperiodicity. Something which should be equivalent to aperiodicity is if

p:=\gcd\{k>0:\nu^{\star k}(e)>0\}

is equal to one (perhaps via invariance \mathbb{P}[\xi_{i+1}=t|\xi_{i}=s]=\mathbb{P}[\xi_{i+1}=th|\xi_{i}=sh]).

If \nu is concentrated on the coset a proper normal subgroup N\rhd G, specifically on Ng\neq Ne, then we have periodicity (Ng\rightarrow Ng^2\rightarrow \cdots\rightarrow Ng^{o(g)}=Ne\rightarrow Ng\rightarrow \cdots), and p=o(g), the order of g.

In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.

The theorem in the quantum case should look like:

Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group G driven by a state \nu\in M_p(G) is ergodic if and only if “X”.

Irreducibility

At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to group-like-projections, which I will say correspond to quasi-subgroups following Kasprzak & Sołtan.

I have shown that if \nu is concentrated on a proper quasi-subgroup S, in the sense that \nu(P_S)=1 for a group-like-projection P_S, that so are the \nu^{\star k}. The analogue of irreducible is that for all q projections in F(G), there exists k\in\mathbb{N} such that \nu^{\star k}(q)>0. If \nu is concentrated on a quasi-subgroup S, then for all k, \nu^{\star k}(Q_S)=0, where Q_S=\mathbf{1}_G -P_S.

I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means

\displaystyle \nu_n:=\frac{1}{n}\sum_{k=1}^n\nu^{\star k},

converge to an idempotent state \nu_\infty. If \nu^{\star k}(q)=0 for all k then the \nu_{\infty}(q)=0 also, so that \nu_\infty\neq \int_G (as the Haar state is faithful). I was able to prove that \nu is supported on the quasi-subgroup given by the idempotent \nu_\infty.

I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.

Aperiodicity

Now onto aperiodicity. Below are propositions which may or may not be true. I am using quantum analogues which may or may not be appropriate. For example, if N\rhd G, then there is the sense of a quotient G/N (which should also take into account a map \pi:F(G)\rightarrow F(N) that describes how N sits inside G. I am taking “concentrated on the coset of a normal subgroup” to be the same as, where P_\nu\in F(G) is the support of \nu, \nu(P_{C_i})=1, where P_{C_{i}} is a minimal projection in F(G/N). I do not know is this appropriate.

The propositions to consider:

  • if \nu is/is not supported on the coset of a proper normal subgroup, then the random walk is periodic/aperiodic.
  • if \nu is/is not supported on the coset of a proper normal quasi-subgroup, then the random walk is periodic/aperiodic.

I do not even have a good definition for periodic… perhaps in the finite case the Haar element \eta\in F(G) can be utilised. Perhaps we might say that the random walk is aperiodic if

p=\gcd\{k>0\,|\,\nu^{\star k}(\eta)>0\}=1.

I have quite a few sheets scrawled at this stage so I might record here some notes and possible approaches. Note that if we have a truly normal quasi-subgroup it may not make sense to reference F(G/N).

– The smallest projection P_\nu such that \nu(P_\nu)=1 is well defined and called in the paper in preparation by the support projection of \nu. Suppose that N\rhd G is a proper normal (quasi?-)subgroup. Let P_{C_i} be a minimal projection in F(G/N). We do not want P_{C_i} to coincide with \mathbf{1}_N=P_N=:P_{C_0} (a group-like-projection in the quasi case). Perhaps to test this we simply require \varepsilon(P_{C_i})=0? As P_{C_i} is positive, we can scale it to make it the density of a state, which I denote by \int_{C_i}. Perhaps the support of \int_{C_i}\star \int_{C_i} is a minimal projection in F(G/N), different to P_{C_i}.

Perhaps I should be using the stochastic operator more. Perhaps T_\nu(P_{C_i}) is a minimal projection if P_\nu\leq P_{C_i}?

– Perhaps even it might be possible to generate periodic behaviour on these \int_{C_i}, that if \left(\int_{C_i}\right)^{\star o(i)}=\int_{C_0}=\int_N, which has support \mathbf{1}_N=P_N, this would make the convolution of powers of \int_{C_i} a cyclic group with identity \int_{C_0}. This would exhibit the periodicity inherent in the classical case and give a necessary condition for aperiodicity, which would be a nice partial result. Let us show a partial result in this direction. Let f be constant the cosets of a normal quasi-subgroup given by a group-like-projection P_N. This should mean, in particular, that

\Delta(f)(\mathbf{1}_G\otimes P_N)=f\otimes P_N.

This means that f\in F(G/N). Now let \nu be supported on a coset C_i so that \nu(P_{C_i})=1 —  where P_{C_i} a minimal projection in F(G/N). Write the above for P_{C_i}:

\Delta(P_{C_i})(\mathbf{1}_G\otimes P_N)=P_{C_i}\otimes P_N.

Hit both sides with \left(\nu\otimes \int_{N}\right) to get:

\sum\nu(P_{C_i(1)})\int_N\left(P_{C_i(2)}P_N\right)=1.

It should be the case, because P_N is the support projection of \int_N, that the left hand side is equal to (\nu\star \int_N)(P_{C_i})=1, that is \nu\star \int_N remains supported on a coset. This mirrors

Ng\cdot Ne=Ng,

for cosets of a classical normal subgroup.

– To chase periodicity we can do other things. Perhaps the period might coincide with

k_T=\min_{T\in\mathbb{N}}\left(P_{\nu^{\star T}}\leq P_N\right).

This would be the time that the random walk would return to the coset Ne. Perhaps in addition we might have

\nu^{\star k}(P_\nu)=0,

for 2\leq k<k_T. Perhaps the simpler \nu^{\star 2}(P_\nu)=0 might be a start.

– We would love to say something like the P_{\nu^{\star k}} are all supported on minimal projections in F(G/N). Might we have P_{C_i}P_{C_j}=0?

– It may be true/useful that for P_\nu the support projection, and \star_A the convolution in F(G):

\displaystyle \frac{P_\nu \star_A P_\nu}{\int_G P_\nu}=P_{\nu\star \nu}.

This might possibly help show that

\displaystyle P_{\nu\star \nu}\leq  \frac{P_{C_i}\star_A P_{C_i}}{\int_G P_{C_i}},

which might possibly be a minimal projection in F(G/N). This seems unlikely and difficult though.

– The theory should be illustratable by looking at the behavior in the Kac-Paljutkin quantum group. Moreover perhaps some periodicity from a truly quasi normal subgroup might be found…

– I am not sure is this useful/true, but in F(G/N), it might be the case that the \int_{C_i} are pure states because the supports are minimal. It might be possible to get periodicity in that way.

– We could try and prove the classical result using the quantum machinery… and see can we write the sufficiency proof using the counit.

– At the end, I must see how this compares to the work of Amaury on C(\widehat{\Gamma})=\mathbb{C}\Gamma, for \Gamma a discrete group.

– This seems as good as place as any to record the work of Franz & Skalski; that a group like projection corresponds to an idempotent:

\displaystyle P_S\longrightarrow \frac{\mathcal{F}(P_S)}{\int_G P_S},

and an idempotent \phi corresponds to a group-like-projection:

\displaystyle \phi\longrightarrow \frac{a_\phi}{\varepsilon(a_\phi)}.

What Now?

What now is that I do not know enough about normal subgroups and cosets to continue. I will now take a study of papers of Podlés and Wang.