## 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. 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.