Large deviations in models of growing clusters with symmetry-breaking transitions

Large deviations in statistics of the convex hull of passive and active particles: A theoretical study

We investigate analytically the distribution tails of the area A and perimeter L of a convex hull for different types of planar random walks. For N noninteracting Brownian motions of duration T we find that the large-L and -A tails behave as P(L)∼e^{-b_{N}L^{2}/DT} and P(A)∼e^{-c_{N}A/DT}, while the small-L and -A tails behave as P(L)∼e^{-d_{N}DT/L^{2}} and P(A)∼e^{-e_{N}DT/A}, where D is the diffusion coefficient. We calculated all of the coefficients (b_{N},c_{N},d_{N},e_{N}) exactly. Strikingly, we find that b_{N} and c_{N} are independent of N for N≥3 and N≥4, respectively. We find that the large-L (A) tails are dominated by a single, most probable realization that attains the desired L (A). The left tails are dominated by the survival probability of the particles inside a circle of appropriate size. For active particles and at long times, we find that large-L and -A tails are given by P(L)∼e^{-TΨ_{N}^{per}(L/T)} and P(A)∼e^{-TΨ_{N}^{area}(sqrt[A]/T)}, respectively. We calculate the rate functions Ψ_{N} exactly and find that they exhibit multiple singularities. We interpret these as DPTs of first order. We extended several of these results to dimensions d>2. Our analytic predictions display excellent agreement with existing results that were obtained from extensive numerical simulations.

Nonequilibrium steady state of trapped active particles

We consider an overdamped particle with a general physical mechanism that creates noisy active movement (e.g., a run-and-tumble particle or active Brownian particle, etc.), that is confined by an external potential. Focusing on the limit in which the correlation time τ of the active noise is small, we find the nonequilibrium steady-state distribution P_{st}(X) of the particle's position X. While typical fluctuations of X follow a Boltzmann distribution with an effective temperature that is not difficult to find, the tails of P_{st}(X) deviate from a Boltzmann behavior: In the limit τ→0, they scale as P_{st}(X)∼e^{-s(X)/τ}. We calculate the large-deviation function s(X) exactly for arbitrary trapping potential and active noise in dimension d=1, by relating it to the rate function that describes large deviations of the position of the same active particle in absence of an external potential at long times. We then extend our results to d>1 assuming rotational symmetry.

Large-deviation theory of increasing returns

An influential theory of increasing returns was proposed by the economist W. B. Arthur in the 1980s to explain the lock-in phenomenon between two competing commercial products. In the most simplified situation there are two competing products that gain customers according to a majority mechanism: each new customer arrives and asks which product they bought to a certain odd number of previous customers, and then buys the most shared product within this sample. It is known that one of these two companies becomes a monopoly almost surely in the limit of infinite customers. Here we consider a generalization [Dosi, Ermoliev, and Kaniovsky, J. Math. Econom. 23, 1 (1994)10.1016/0304-4068(94)90032-9] in which the new customer follows the indication of the sample with some probability, and buys the other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane, and Sudderth, and it includes several models of physical interest as special cases, such as the Elephant Random Walk, Friedman's urn, and other generalized urn models. We provide a large-deviation analysis of this model at the sample-path level, and we provide a formula that allows us to find the most likely trajectories followed by the market share variable. Interestingly, in the parameter range where the lock-in phase is expected, we observe a whole region of convergence where the entropy cost is sublinear. We also find a nonlinear differential equation for the cumulant-generating function of the market share variable, which can be studied with a suitable perturbation theory.

Varied phenomenology of models displaying dynamical large-deviation singularities

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory

We analyze large deviations of time-averaged quantities in stochastic processes with long-range memory, where the dynamics at time t depends itself on the value q_{t} of the time-averaged quantity. First we consider the elephant random walk and a Gaussian variant of this model, identifying two mechanisms for unusual fluctuation behavior, which differ from the Markovian case. In particular, the memory can lead to large-deviation principles with reduced speeds and to nonanalytic rate functions. We then explain how the mechanisms operating in these two models are generic for memory-dependent dynamics and show other examples including a non-Markovian simple exclusion process.