Ergodic properties of a random shuffling process

Question

Consider the following continuous analogue of a card shuffling process:

Let $Y_i, Z_i$ ($i \in \mathbb Z^+$) be sequences of jointly independent uniformly distributed random variables on $[0, 1]$. Denote by $M_i =: [a_i, b_i]$ the closed interval having $Y_i$ and $Z_i$ as endpoints.

For each $n \in \mathbb Z_+$, let $T_n: [0, 1] \to [0, 1]$ be the (random) map defined by

$$ T_n (x) = \begin{cases} x + b_n - a_n& \text{if }\; x < a_n,\\ x - a_n & \text{if }\; a_ i \leq x \leq b_n,\\ x, &\text{if }\; x > b_n. \end{cases} $$

Thus each $T_n$ takes a random segment from the middle of the deck $[0, 1]$ and places it at the top of the deck.

It is immediate that $T_n$ are measure preserving with respect to the Lebesgue measure, almost surely.

Question: Is it true that $T_n$ are almost surely strong mixing?

We say that $T_n$ are strongly mixing if almost surely,

$$\lim_{n \to \infty} \mu( T_n \dots T_1 (A) \cap B) = \mu(A)\mu(B)$$

for all Borel sets $A$, $B$.

Answer 1

Here is an approach that can prove an average weak mixing of this system, ie:

$$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \mathbb{E} (\mu( T_k \dots T_1 (A) \cap B)) = \mu(A)\mu(B)$$

for all Borel sets $A$, $B$.

By a standard approximation argument, this is equivalent to:

$$ \lim_{n \to \infty} \mathbb{E}( \int (f \circ T_1 \circ \cdots \circ T_n) \cdot g ) = \int f \cdot g , $$ for all $f,g \in L^2([0,1])$.

We study the averaging operator $P$: $$ Pf = \mathbb{E}( f \circ T ) . $$ The operator $P$ is self-adjoint on $L^2([0,1])$, with operator norm 1, and $$ P^n f = \mathbb{E}( f \circ T_1 \circ \cdots \circ T_n ) . $$ By (an adaptation of) Von Neumann mean ergodic theorem we have, for every $f \in L^2([0,1])$, the convergence in $L^2$: $$ \frac{1}{n} \sum_{k=1}^n P^n f \to \pi(f) , $$ where $\pi$ is the orthogonal projection on the closed subspace of $P$-invariant function. In particular, the average weak mixing property above is equivalent to the non-existence of non-constant $P$-invariant function.

We can compute $P$ explicitly: $$ Pf(x) = 2 \int_x^1 \int_0^t f(u)dudt + x^2f(x) . $$ This reduces the question of average weak mixing to the 2nd order integral equation: $$ f(x) = 2 \int_x^1 \int_0^t f(u)dudt + x^2f(x) . $$ I found that there is no non-constant $L^2$ solution to this equation, by reducing to a differential equation (this should be verified !). This proves the average weak mixing.

I guess that the average mixing $$\lim_{n \to \infty} \mathbb{E} (\mu( T_n \dots T_1 (A) \cap B)) = \mu(A)\mu(B)$$ could be deduced (if true) from a spectral gap of $P$ on some appropriate function space.