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Register a new userPhase transitions for degenerate random environments
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We study a class of models of i.i.d.~random environments in general dimensions $dge 2$, where each site is equipped randomly with an environment, and a parameter $p$ governs the frequency of certain environments that can act as a barrier. We show that many of these models (including some which are non-monotone in $p$) exhibit a sharp phase transition for the geometry of connected clusters as $p$ varies.
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We study continuous-time (variable speed) random walks in random environments on $mathbb{Z}^d$, $dge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic with respect to space-time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer-Sjostrand representation of gradient models with certain non-strictly convex potentials.
We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barabasi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barabasi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors.
Products of $M$ i.i.d. non-Hermitian random matrices of size $N times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix theory (finite $M$ and large $N$). The remaining task is to study local eigenvalue statistics as $M$ and $N$ tend to infinity simultaneously, which lies at the heart of understanding two kinds of universal patterns. For products of i.i.d. complex Ginibre matrices, truncated unitary matrices and spherical ensembles, as $M+Nto infty$ we prove that local statistics undergoes a transition when the relative ratio $M/N$ changes from $0$ to $infty$: Ginibre statistics when $M/N to 0$, normality when $M/Nto infty$, and new critical phenomena when $M/Nto gamma in (0, infty)$.
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $mathcal{P}_{lambda}$ in $mathbb{R}^2$ of intensity $lambda$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(|x-y|)$, independent of everything else, where $g:[0,infty) to [0,1]$ and $| cdot |$ is the Euclidean norm. In the inhomogenous version of the model, points of $mathcal{P}_{lambda}$ are endowed with weights that are non-negative independent random variables with distribution $P(W>w)= w^{-beta}1_{[1,infty)}(w)$, $beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability $1 - expleft( - frac{eta W_xW_y}{|x-y|^{alpha}} right)$, $eta, alpha > 0$, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of $mathcal{P}_{lambda}$. A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of $mathcal{P}_{lambda}$. Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.
We first study a model, introduced recently in cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle placed there. The obstacles appear at each site with probability $pin [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $mathbb P_p$-measure, as $ntoinfty$, (i) Critical case: P^{omega}(S_n)simfrac{2}{qn}; (ii) Subcritical case: P^{omega}(S_n)= expleft[left( -C_{d,q}cdot frac{n}{(log n)^{2/d}} right)(1+o(1))right], where $C_{d,q}>0$ does not depend on the branching law. Hence, the model exhibits `self-averaging in the critical case but not in the subcritical one. I.e., in (i) the asymptotic tail behavior is the same as in a toy model where space is removed, while in (ii) the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. We utilize a spine decomposition of the branching process as well as some known results on random walks.
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