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Rubik and Markov. Right here we get hold of the likelihood of… | by Eduardo Testé | Aug, 2023

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A Markov course of on depth’s lessons

To search out p(d|N) we think about the depth lessons as websites of a Markov course of. Let me clarify:

Randomly shifting the dice faces is described as a Markov course of (one dimensional random stroll) between depth lessons. Picture by the creator.

A depth class d is the set of all of the dice’s states at a depth d (minimal variety of strikes to the solved state). If we randomly selected a state in a depth class d, and we flip a random face with a random transfer, that can give us both a state within the class d + 1 , with a likelihood p_d, or a state within the class d -1, with a likelihood q_d. Within the quarter-turn metric there are not any self-class strikes.

That defines a Markov course of, the place a selected website is an entire depth class. In our case, solely contiguous d lessons are one-jump linked. To be extra exact, this can be a discrete-time birth-death Markov chain. As a result of the quantity of web sites is finite, the chain can also be irreducible (ergodic), and a novel stationary distribution exist.

We assume equally distributed possibilities for the number of the random strikes at every time. That induces some transition possibilities p_d, q_d (to be computed) between the depth lessons. The quantity of random strikes N is the discrete time of the Markov course of. That is additionally a one-dimensional random walker: at each website (depth class quantity d), the likelihood of going ahead is p_d, and the likelihood of going backwards is q_d. This one dimensional chain is, roughly talking, the “radial” path within the Rubik’s graph (organized within the depth-radial structure).

The transition matrix

Any Markov processes is encoded in a transition matrix M. The (i,j) entry of M is the likelihood of leaping from website i to website j. In our case solely the next entries are completely different from zero:

Right here p_0 = 1: from the depth class 0 (containing simply the solved state) we will solely leap to the depth class 1 (there isn’t any class -1). Additionally, q_26 = 1: from the depth class 26 we will solely leap to depth class 25 (there isn’t any class 27). For a similar motive, there are not any p_26 or q_0.

The stationary distribution

We mapped the motion of randomly shifting the dice to a one-dimensional depth-class random walker leaping backwards and forwards with possibilities q_d and p_d. What occurs after an extended stroll? or, what number of instances does the walker go to a selected website after an extended stroll? In actual life: how usually is a depth class visited when the dice undergoes random turns?

In the long term, and it doesn’t matter what the place to begin was, the time the walker spends within the depth class d is proportional the inhabitants D(d) of that depth class. That is the primary level right here:

the (normalized) depth-population record D(d) needs to be interpreted because the vector representing the stationary distribution of our depth class Markov course of.

Mathematically, D(d) is a left eigenvector of M

This matrix equation will give us 26 linear equations, from which we’ll get the p_i’s and q_is.

Bearing in mind that p_0 = q_26 = 1, we will rewrite these as

Detailed steadiness equations. Picture by the creator.

These are referred to as detailed steadiness equations: the flux, outlined to be the stationary website inhabitants instances the leaping likelihood, is similar in each instructions. The options are:

and p_i is obtained utilizing p_i + q_i = 1.

Some circumstances on the inhabitants of a depth class

There’s something fascinating about these options. As a result of q_i is a likelihood we should always have that

and that translate into the next situation for the distribution D_k:

This can be a tower of inequalities that the depth-population D_k ought to fulfill. Explicitly, they are often organized as:

Particularly, the final two inequalities are

As a result of D_27 = 0, we get that the decrease and higher sure are equal, so

Or:

The sum of the inhabitants of the even websites needs to be equal to the sum of the inhabitants of the odd websites!

We are able to see this as an in depth steadiness between even and odd websites: each transfer is all the time to a unique and contiguous depth class. Any leap will take you from the odd depth class (the category of all of the odd depth lessons) to the even depth class (the category of all of the even depth lessons). So the odd to even class leap happen with likelihood 1 (and vise versa). Being the possibilities one in each path, their inhabitants needs to be equal by detailed steadiness.

For a similar motive the Markov course of will attain a period-two “stationary distribution” that switches between even and odd websites after each transfer (discrete time N).

An issue with the info

The depth-population D_d reported within the supply of the info we’re planning to make use of is approximate for d = 19,20,21,22,23,24. So there isn’t any assure it can fulfill all these circumstances (inequalities). Don’t be stunned if we get some possibilities q_i out of the vary [0,1] (as it’s the case!). Particularly, if we try to verify the final situation (the even-odd inhabitants equality) it’s off by an enormous quantity! (replace: see be aware on the finish)

A method out

The odd class appear to be underpopulated (this can be a consequence of the chosen approximation to report the info). To make issues work (get possibilities within the vary [0,1]), we determine so as to add the earlier huge quantity to the inhabitants of the depth class 21 (the odd class with the best inhabitants, or, the one that can discover that addition the least). With this correction, all of the obtained possibilities appears to be appropriate (which suggests the inequalities are additionally happy).

The leaping possibilities are:

p_i = 
{1., 0.916667, 0.903509, 0.903558, 0.903606, 0.903602, 0.90352, 0.903415,
0.903342, 0.903292, 0.903254, 0.903221, 0.903189, 0.903153, 0.903108,
0.903038, 0.902885, 0.902409, 0.900342, 0.889537, 0.818371, 0.367158,
0.00342857, 6.24863*1e-12, 0.00022, 0.0833333}
# i from 0 to 25

q_i =
{0.0833333, 0.0964912, 0.0964419, 0.096394, 0.0963981, 0.0964796,
0.096585, 0.096658, 0.0967081, 0.0967456, 0.0967786, 0.0968113,
0.0968467, 0.0968917, 0.0969625, 0.0971149, 0.0975908, 0.0996581,
0.110463, 0.181629, 0.632842, 0.996571, 1., 0.99978, 0.916667, 1.}
# i from 1 to 26

Discover that the majority the primary p_i (as much as i = 21) are near 1. These are the possibilities of going away from the solved state. The chances of going nearer to the solved state (q_i) are virtually 1 for i better than 21. This places in perspective why it’s tough to unravel the dice: the random walker (or the dice’s random mover) can be “trapped perpetually” in a neighborhood of the depth class 21.



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