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The Erdős-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if LN is
the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then
π LN converges in distribution to an exponential random variable with parameter one. This log N
can be equivalently stated in terms of the total cřollision time of two independent simple random walks on the plane. More precisely, if Lp1,2q " N 1 p1q p2q , then π Lp1,2q converges in
N n"1 tSn"Snu logN N
distribution to an exponential random variable of parameter one. We prove that for every h ě
3, the family ␣ π Lpi,jq( , of logarithmically rescaled, two-body collision local times log N N 1ďiăjďh
between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős-Taylor theorem. We also discuss connections to directed polymers in random environments.
If given the choice, what strategy should agents use to switch partners in strategic social interactions? While many analyses have been performed on specific switching heuristics, showing how and when these lead to more cooperation, no insights have been provided into which rule will actually be learnt by agents when given the freedom to do so. Starting from a baseline model that has demonstrated the potential of rewiring for cooperation, we provide answers to this question over the full spectrum of social dilemmas. Multi-agent Q-learning with Boltzmann exploration is used to learn when to sever or maintain an association. In both the Prisoner's Dilemma and the Stag Hunt games we observe that the Out-for-Tat rewiring rule, breaking ties with other agents choosing socially undesirable actions, becomes dominant, confirming at the same time that cooperation flourishes when rewiring is fast enough relative to imitation. Nonetheless, in the transitory region before full cooperation, a Stay strategy, keeping a connection at all costs, remains present, which shows that loyalty needs to be overcome for full cooperation to emerge. In conclusion, individuals learn cooperation-promoting rewiring rules but need to overcome a kind of loyalty to achieve full cooperation in the full spectrum of social dilemmas.
The critical 2d Stochastic Heat Flow (SHF) is a stochastic process of random measures on ℝ2, recently constructed in [CSZ23]. We show that this process falls outside the class of Gaussian Multiplicative Chaos (GMC), in the sense that it cannot be realised as the exponential of a (generalised) Gaussian field. We achieve this by deriving strict lower bounds on the moments of the SHF that are of independent interest.
We consider directed polymers in random environment in the critical dimension d=2, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on R2 with logarithmic correlations, which we call the *Critical 2d Stochastic Heat Flow*. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.
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