Imagine a ball in an unfamiliar hall: two guests start from opposite corners. Their movements are not a smooth waltz but chaotic improvisation in search of rare treasures. Contrary to intuition, the outcome hinges not on tempo but on the geometry of the floor. This scenario, echoing Albert Einstein’s ideas on the Brownian dance of particles and Andrey Markov’s chains of random events, now finds a universal conclusion.

The key to the choreography is the spectral dimension $d_s$. It dictates how quickly the random dance ‘forgets’ its opening step. In environments with $d_s$ < 2—such as fractal mazes or porous mirrors — the memory of the start never fades. Here shares never average out: inequality freezes, as if quantum entanglement forever binds the partners with an invisible thread. An unseen dark matter of correlations pervades the process, and collapse toward equal division never occurs.

For 2 ≤ $d_s$ < 3, the dance slows: equalization proceeds, but at a snail’s pace, obeying non-Gaussian laws. Only when $d_s$ ≥ 3 does the hall become ‘spacious’—fluctuations take on a Gaussian character, and the entropy of resource distribution grows predictably. This universality has been verified by numerical simulations for ordinary Brownian motion, superdiffusion, and even memory systems.

Curiously, the phase transition at $d_s$ = 2 and 3 recalls the Big Bang — here a new statistics is born. According to the standard model of random processes, this is an intrinsic property of geometry, independent of the dance’s details.

The finding introduces an uncertainty principle into forecasting competitive systems: even with exact symmetry of rules, the outcome can be arbitrarily unfair. Microscopic symmetry does not guarantee macroscopic equality: this principle changes our view of fair competition. It echoes the expansion of the universe — some inequalities, once born, only grow in scale. Going forward, scientists plan to study unequal opponents and environmental interactions, like the pheromone trails of ants. Such models could rewrite scenarios in ecology, economics, and distributed computing.