Researcher:
Yılmaz, Atilla

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Atilla

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Yılmaz

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Yılmaz, Atilla

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2015 - 20196

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    The stochastic encounter-mating model
    (Springer, 2017) Guen, Onur; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We propose a new model of permanent monogamous pair formation in zoological populations with multiple types of females and males. According to this model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, we analyze the contingency table of permanent pair types in three cases: (i) definite mating upon encounter; (ii) Poisson firing times; and (iii) Bernoulli firing times. In the first case, the contingency table has a multiple hypergeometric distribution which implies panmixia. The other two cases generalize the encounter-mating models of Gimelfarb (Am. Nat. 131(6):865-884, 1988) who gives conditions that he conjectures to be sufficient for panmixia. We formulate adaptations of his conditions and prove that they not only characterize panmixia but also allow us to reduce the model to the first case by changing its underlying parameters. Finally, when there are only two types of females and males, we provide a full characterization of panmixia, homogamy and heterogamy.
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    Nonconvex homogenization for one-dimensional controlled random walks in random potential
    (2019) Zeitouni, Ofer; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences; N/A
    We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk {Xi} on the set of integers. The cost function is the expectation of the exponential of the path sum of a random stationary and ergodic bounded potential plus θXn. The random walk policies are measurable with respect to the random potential, and are adapted, with their drifts uniformly bounded in magnitude by a parameter δ∈[0,1]. Under natural conditions on the potential, we prove that the normalized logarithm of the optimal cost function converges. The proof is constructive in the sense that we identify asymptotically optimal policies given the value of the parameter δ, as well as the law of the potential. It relies on correctors from large deviation theory as opposed to arguments based on subadditivity which do not seem to work except when δ=0. The Bellman equation associated to this control problem is a second-order Hamilton–Jacobi (HJ) partial difference equation with a separable random Hamiltonian which is nonconvex in θ unless δ=0. We prove that this equation homogenizes under linear initial data to a first-order HJ equation with a deterministic effective Hamiltonian. When δ=0, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in θ. In contrast, when δ=1, the effective Hamiltonian is piecewise linear and nonconvex in θ. Finally, when δ∈ (0,1), the effective Hamiltonian is expressed completely in terms of the tilted free energy for the δ=0 case and its convexity/nonconvexity in θ is characterized by a simple inequality involving δ and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.
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    Fluid limit for the Poisson encounter-mating model
    (Applied Probability Trust, 2017) Gün, Onur; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences
    Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this paper we study SEM models with Poisson firing times. First, we prove that the model enjoys a fluid limit as the population size diverges, that is, the stochastic dynamics converges to a deterministic system governed by coupled ordinary differential equations (ODEs). Then we convert these ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Next, under the so-called fine balance condition which characterizes panmixia, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of female and male. Without the fine balance assumption, but under certain symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to characterize assortative mating.
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    PublicationOpen Access
    Variational formulas and disorder regimes of random walks in random potentials
    (International Statistical Institute (ISI), 2017) Rassoul-Agha, Firas; Seppalainen, Timo; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences; 26605
    We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder.
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    Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment
    (University of Washington Press, 2017) Rassoul-Agha, Firas; Seppalainen, Timo; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences; 26605
    We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears.
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    Ratios of partition functions for the log-gamma polymer
    (Institute of Mathematical Statistics (IMS), 2015) Georgiou, Nicos; Rassoul-Agha, Firas; Seppaelaeinen, Timo; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences; 26605
    We introduce a random walk in random environment associated to an underlying directed polymer model in 1 + 1 dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment.