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Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first show that…

We study the total mass of the solution to the parabolic Anderson model on a regular tree with an i.i.d. random potential whose marginal distribution is double-exponential. In earlier work we identified two terms in the asymptotic expansion…

This paper investigates contraction properties of switched dynamical systems for the case that all modes are non-contracting, thereby extending existing results that require at least one mode to be contracting. Leveraging the property that…

We considered diffusion-driven processes on small-world networks with distance-dependent random links. The study of diffusion on such networks is motivated by transport on randomly folded polymer chains, synchronization problems in…

We use the Quench Action approach to study the non-equilibrium dynamics after a quantum quench in the Hubbard model in the limit of infinite interaction. We identify a variety of low-entangled initial states for which we can directly…

We explore analytically the oscillation quenching phenomena (amplitude death and oscillation death) in a coupled third order phase locked loop (PLL) both in periodic and chaotic mode. The phase locked loops are coupled through mean field…

We propose a family of lagged random walk sampling methods in simple undirected graphs, where transition to the next state (i.e. node) depends on both the current and previous states -- hence, lagged. The existing random walk sampling…

We prove a random Ruelle--Perron--Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions,…

The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q=…

The limiting distribution of eigenvalues of N x N random matrices has many applications. One of the most studied ensembles are real symmetric matrices with independent entries iidrv; the limiting rescaled spectral measure (LRSM)…

We consider the small deviation probability for random walk with time-inhomogeneous random environment. Compared with the result in Mogul'ski\u{\i} (1974) for the i.i.d. random walk, the rate is smaller (due to the random environment),…

Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Voln\'{y} (2018) showed that the quenched CLT (central limit theorem) holds for ortho-martingale random fields. In this paper, we study the…

We formulate a criterion for the existence of an invariant measure for a Feller semigroup defined on a metric space with the e-property for bounded continuous functions and use it to prove the asymptotic stability of a semigroup satisfying…

We introduce a model of a quantum walk on a graph in which a particle jumps between neighboring nodes and interacts with independent spins sitting on the edges. Entanglement propagates with the walker. We apply this model to the case of a…

Disordered systems are interesting for many physical reasons. In this article, we study the renormalization group property of quenched disorder systems in the presence of a boundary. We construct examples of scalar field theories in various…

This thesis investigates critical phenomena and equilibrium states in various stochastic models through three interconnected studies. In the first chapter, we analyze the Activated Random Walk model on a one-dimensional ring in the…

In the first part of the article our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which in turn are defined by a sequence of…

We consider quantum quenches in the integrable $SU(3)$-invariant spin chain (Lai-Sutherland model) which admits a Bethe ansatz description in terms of two different quasiparticle species, providing a prototypical example of a model solvable…

The study of quantum quenches in integrable systems has significantly advanced with the introduction of the Quench Action method, a versatile analytical approach to non-equilibrium dynamics. However, its application is limited to those…

The renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the Gaussian self-similar velocity field with finite, and not small, correlation time. The inertial-range energy…