It Began from a Random Encounter

"Why did you join the information theory club?"

At the new member briefing, Aoi asked Yuki.

"By chance. I saw you in the library, senpai, and the book title you held looked interesting."

"A random encounter."

"But," Yuki continued, "various things happened from there, and now I'm here. It doesn't feel entirely random."

Aoi nodded with interest. "There's a concept called stochastic processes."

"Stochastic processes?"

Riku interrupted. "Mine was chance too! I entered the wrong room."

"In your case, it's chaotic," Aoi laughed.

Mira quietly approached and wrote in her notebook. "Random walk leads to destination"

"As Mira says," Aoi began explaining, "even a series of random choices eventually converges to a certain state. This is a Markov chain."

Yuki asked, "Markov?"

"A stochastic process where the next state is determined only from the current state. Past paths don't matter."

"So the probability of me being here now can be calculated?"

Aoi drew a diagram on the whiteboard.

"For example, consider Yuki's daily states. Library, classroom, club room, home. There are transition probabilities from each state to the next."

"Like 0.3 probability of going from library to club room, 0.7 to home?"

"Yes. And what's interesting is that after sufficient time, it converges to a stationary distribution."

Riku looked confused. "Stationary distribution?"

"The long-term proportion of time spent in each state. Independent of the initial state."

Mira wrote an equation. "π = πP"

"The stationary distribution π equals its product with the transition matrix P," Aoi supplemented.

Yuki pondered. "So the probability of me being in the club room also becomes constant long-term?"

"For an ergodic Markov chain, yes. When you can reach any state from any other state."

Riku said, "So us meeting was also a result of a Markov chain?"

"In a sense. But," Aoi smiled, "human behavior isn't completely Markovian. Past experiences influence us too."

Mira wrote a new note. "Memory makes us more than Markov"

"Memory makes us more than Markov," Yuki translated.

Aoi nodded. "Yes. Information theory often assumes memorylessness, but reality is more complex."

Yuki looked out the window. "But it's true it began from a random encounter."

"The probability might have been low," Aoi said. "But it wasn't zero. So it happened."

Riku raised his hand. "So whether I'm late tomorrow is also a Markov chain?"

"In simplified form. Whether you were late today alone affects tomorrow's transition probability."

"So being late yesterday makes me more likely to be late today?"

"That's a process with memory. Not Markov, but a higher-order Markov model."

Mira smiled quietly. She enjoyed these discussions.

Yuki said, "Even appearing random, there's a probability structure behind it."

"Yes. The PageRank algorithm is also a Markov chain. It evaluates random movement between web pages with stationary distribution."

"Information theory really applies everywhere."

Aoi nodded. "Probability, statistics, information theory. They're all connected."

Riku's face became serious. "So what's the probability of us four gathering?"

"To calculate, we'd need all transition probabilities," Aoi answered. "But it's certainly very low."

"That's why," Yuki smiled, "this encounter has value."

Mira nodded. Her expression was calmer than usual.

Sunset streamed through the club room window. Random encounters had become certain bonds.

Markov chains sometimes create miracles.