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We give a complete and unified description -- under some stability assumptions -- of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we…
Motivated by understanding the phase structure of $d >1$ strings we investigate the $c=1$ matrix model with $g' (\tr M(t)^{2})^{2}$ interaction which is the simplest approximation of the model expected to describe the critical phenomena of…
We study two types of stochastic processes, a mean-field spatial system of interacting Fisher-Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and a (mean-field) spatial system of…
The extraordinary transition which occurs in the two-dimensional O(n) model for $n<1$ at sufficiently enhanced surface couplings is studied by conformal perturbation theory about infinite coupling and by finite-size scaling of the spectrum…
An investigation of the spatial fluctuations and their manifestations in the vicinity of the quantum critical point within the framework of the renormalized $\phi^{4}$ theory is proposed. Relevant features are reported through the…
Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest; from the control of epidemics to understanding brain dynamics. While in some cases there exists a…
Motivated by applications to COVID dynamics, we describe a branching process in random environments model $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving…
We investigate the critical behavior of three-dimensional relativistic fermion models with a U(N_L)_L x U(1)_R chiral symmetry reminiscent of the Higgs-Yukawa sector of the standard model of particle physics. We classify all possible…
In the present paper, we characterize the behavior of supercritical branching processes in random environment with linear fractional offspring distributions, conditioned on having small, but positive values at some large generation. As it…
We study branching processes in an i.i.d. random environment, where the associated random walk is of the oscillating type. This class of processes generalizes the classical notion of criticality. The main properties of such branching…
We study non-equilibrium analogues of surface phase transitions in a minimal model of active particles in contact with a purely repulsive potential barrier that mimics a thin porous membrane. Under conditions of bulk motility-induced phase…
Interacting fermions exhibit a rich landscape of surface defects and associated critical phenomena. We investigate novel surface critical behavior in the three-dimensional Gross-Neveu-Yukawa model. For a class of defect renormalization…
The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bm{\phi}$ and short-range interactions is investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an…
We study a continuous quasi-two-dimensional order-disorder phase transition that occurs in a simple model of a material that is inhomogeneously strained due to the presence of dislocation lines. Performing Monte Carlo simulations of…
We introduce and numerically study the branching annihilating random walks with long-range attraction (BAWL). The long-range attraction makes hopping biased in such a manner that particle's hopping along the direction to the nearest…
Using field theoretic renormalization group methods we study the critical behavior of a driven diffusive system near a boundary perpendicular to the driving force. The boundary acts as a particle reservoir which is necessary to maintain the…
We study the appearance of first-order dynamical phase transitions (DPTs) as `intermittent' co-existing phases in the fluctuations of random walks on graphs. We show that the diverging time scale leading to critical behaviour is the waiting…
We study the scaling properties of long-range loop-erased random walks (LR-LERW), where the underlying random walker performs L\'evy-flight-like jumps with a power-law step-length distribution $P(\mathbf{r})\sim |\mathbf{r}|^{-(d+\sigma)}$.…
A cellular automaton model is presented for random walkers with biologically motivated interactions favoring local alignment and leading to collective motion or swarming behavior. The degree of alignment is controlled by a sensitivity…
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…