Tobias Galla

The supersymmetric Ward identities on the lattice

Abstract Supersymmetric (SUSY) Ward identities are considered for the N=1 SU(2) SUSY Yang-Mills theory discretized on the lattice with Wilson fermions (gluinos). They are used in order to compute non-perturbatively a subtracted gluino mass and the mixing coefficient of the SUSY current. The computations were performed at gauge coupling

Complex dynamics in learning complicated games

\beta=2.3

How limit cycles and quasi-cycles are related in systems with intrinsic noise

and hopping parameter

Intrinsic noise in game dynamical learning.

\kappa\!=\!0.1925

Limit cycles, complex Floquet multipliers, and intrinsic noise

, 0.194, 0.1955 using the two-step multi-bosonic dynamical-fermion algorithm. Our results are consistent with a scenario where the Ward identities are satisfied up to O(a) effects. The vanishing of the gluino mass occurs at a value of the hopping parameter which is not fully consistent with the estimate based on the chiral phase transition. This suggests that, although SUSY restoration appears to occur close to the continuum limit of the lattice theory, the results are still affected by significant systematic effects.

Evolutionary dynamics, intrinsic noise, and cycles of cooperation

Game theory is the standard tool used to model strategic interactions in evolutionary biology and social science. Traditionally, game theory studies the equilibria of simple games. However, is this useful if the game is complicated, and if not, what is? We define a complicated game as one with many possible moves, and therefore many possible payoffs conditional on those moves. We investigate two-person games in which the players learn based on a type of reinforcement learning called experience-weighted attraction (EWA). By generating games at random, we characterize the learning dynamics under EWA and show that there are three clearly separated regimes: (i) convergence to a unique fixed point, (ii) a huge multiplicity of stable fixed points, and (iii) chaotic behavior. In case (iii), the dimension of the chaotic attractors can be very high, implying that the learning dynamics are effectively random. In the chaotic regime, the total payoffs fluctuate intermittently, showing bursts of rapid change punctuated by periods of quiescence, with heavy tails similar to what is observed in fluid turbulence and financial markets. Our results suggest that, at least for some learning algorithms, there is a large parameter regime for which complicated strategic interactions generate inherently unpredictable behavior that is best described in the language of dynamical systems theory.

Effects of noise and confidence thresholds in nominal and metric Axelrod dynamics of social influence.

Fluctuations and noise may alter the behaviour of dynamical systems considerably. For example, oscillations may be sustained by demographic fluctuations in biological systems where a stable fixed point is found in the absence of noise. We here extend the theoretical analysis of such stochastic effects to models which have a limit cycle for some range of the model parameters. We formulate a description of fluctuations about the periodic orbit which allows the relation between the stochastic oscillations in the fixed-point phase and the oscillations in the limit cycle phase to be elucidated. In the case of the limit cycle, a suitable transformation into a co-moving frame allows fluctuations transverse and longitudinal with respect to the limit cycle to be effectively decoupled. While longitudinal fluctuations are of a diffusive nature, those in the transverse direction follow a stochastic path more akin to that of an Ornstein-Uhlenbeck process. Their power spectrum is computed analytically within a van Kampen expansion in the inverse system size. The subsequent comparison with numerical simulations, carried out in two different ways, illustrates the effects that can occur due to diffusion in the longitudinal direction.

Intrinsic fluctuations in stochastic delay systems: Theoretical description and application to a simple model of gene regulation

Demographic noise has profound effects on evolutionary and population dynamics, as well as on chemical reaction systems and models of epidemiology. Such noise is intrinsic and due to the discreteness of the dynamics in finite populations. We here show that similar noise-sustained trajectories arise in game dynamical learning, where the stochasticity has a different origin: agents sample a finite number of moves of their opponents in between adaptation events. The limit of infinite batches results in deterministic modified replicator equations, whereas finite sampling leads to a stochastic dynamics. The characteristics of these fluctuations can be computed analytically using methods from statistical physics, and such noise can affect the attractors significantly, leading to noise-sustained cycling or removing periodic orbits of the standard replicator dynamics.

Fixation in finite populations evolving in fluctuating environments

We study the effects of intrinsic noise on chemical reaction systems, which in the deterministic limit approach a limit cycle in an oscillatory manner. Previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude. We show that a similar effect occurs when the stable attractors are limit cycles. We compute the correlation functions and spectral properties of the fluctuations in suitably comoving Frenet frames for several model systems including driven and coupled Brusselators, and the Willamowski-Rössler system. Analytical results are confirmed convincingly in numerical simulations. The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.

microRNA input into a neural ultradian oscillator controls emergence and timing of alternative cell states

We use analytical techniques based on an expansion in the inverse system size to study the stochastic evolutionary dynamics of finite populations of players interacting in a repeated prisoners dilemma game. We show that a mechanism of amplification of demographic noise can give rise to coherent oscillations in parameter regimes where deterministic descriptions converge to fixed points with complex eigenvalues. These quasicycles between cooperation and defection have previously been observed in computer simulations; here we provide a systematic and comprehensive analytical characterization of their properties. We are able to predict their power spectra as a function of the mutation rate and other model parameters and to compare the relative magnitude of the cycles induced by different types of underlying microscopic dynamics. We also extend our analysis to the iterated prisoners dilemma game with a win-stay lose-shift strategy, appropriate in situations where players are subject to errors of the trembling-hand type.