Related papers: Fluctuations of random Motzkin paths
In this paper we discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to $\infty $.…
We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current…
We consider a standard one-dimensional Brownian motion on the time interval $[0,1]$ conditioned to have vanishing iterated time integrals up to order $N$. We show that the resulting processes can be expressed explicitly in terms of shifted…
The evolution of many stochastic systems is accurately described by random walks on graphs. We here explore the close connection between local steady-state fluctuations of random walks and the global structure of the underlying graph.…
We compute limit fluctuations of random Motzkin paths with arbitrary end-points as the length of the path tends to infinity.
Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${\mathbb Z}$. It is a non-Markovian process where the steps form a (possibly)…
Many studies assume stock prices follow a random process known as geometric Brownian motion. Although approximately correct, this model fails to explain the frequent occurrence of extreme price movements, such as stock market crashes. Using…
We examine a new path transform on 1-dimensional simple random walks and Brownian motion, the quantile transform. This transformation relates to identities in fluctuation theory due to Wendel, Port, Dassios and others, and to discrete and…
We study a Brownian excursion on the time interval $\left|t\right|\leq T$, conditioned to stay above a moving wall $x_{0}\left(t\right)$ such that $x_0\left(-T\right)=x_0\left(T\right)=0$, and $x_{0}\left(\left|t\right|<T\right)>0$. For a…
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is…
Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant $SC_n(t)$ matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics…
We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal…
For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the…
We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random…
We present a new approach to fluctuation identities for reflected L\'{e}vy processes with one-sided jumps. This approach is based on a number of easy to understand observations and does not involve excursion theory or It\^{o} calculus. It…
We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on $\mathbb Z^d$, $d\ge 2$. Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has…
We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and…
In this note - starting from $d$-dimensional (with $d>1$) fuzzy vectors - we prove Donsker's classical invariance principle. We consider a fuzzy random walk ${S^*_n}=X^*_1+\cdots+X^*_n,$ where $\{X^*_i\}_1^{\infty}$ is a sequence of…
Understanding fluctuations of observables across stochastic trajectories is essential for various fields of research, from quantum thermal machines to biological motors. We introduce a framework to analyze the statistics of counting…