arXiv subjects

Hugo Panzo

Publications and source records attributed to Hugo Panzo.

21 records · Page 2Linked to original sources

arXiv2018

The subcritical phase for a homopolymer model

We study a model of continuous-time nearest-neighbor random walk on $\mathbb{Z}^d$ penalized by its occupation time at the origin, also known as a homopolymer. For a fixed real parameter $β$ and time $t>0$, we consider the probability measure on paths of the random walk starting from the origin whose Radon-Nikodym derivative is proportional to the exponent of the product $β$ times the occupation time at the origin up to time $t$. The case $β>0$ was studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the case $β<0$, which is intrinsically different only when the underlying walk is recurrent, that is $d=1,2$. Our main result is a scaling limit for the distribution of the homopolymer on the time interval $[0,t]$, as $t\to\infty$, a result that coincides with the scaling limit for penalized Brownian motion due to Roynette and Yor. In two dimensions, the penalizing effect is asymptotically diminished, and the homopolymer scales to standard Brownian motion. Our approach is based on potential analytic and martingale approximation for the model. We also apply our main result to recover a scaling limit for a wetting model. We study the model through analysis of resolvents.

math.PR↗

arXiv2014

Efficient Coupling for Random Walk with Redistribution

What can one say on convergence to stationarity of a finite state Markov chain that behaves "locally" like a nearest neighbor random walk on ${\mathbb Z}$ ? The model we consider is a version of nearest neighbor lazy random walk on the state space $ \{0,\dots,N\}$: the probability for staying put at each site is $\frac 12$, the transition to the nearest neighbors, one on the right and one on the left, occurs with probability $\frac14$ each, where we identify two sites, $J_0$ and $J_N$ as, respectively, the neighbor of $0$ from the left and the neighbor of $N$ from the right (but $0$ is not a neighbor of $J_0$ and $N$ is not neighbor of $J_N$). This model is a discrete version of diffusion with redistribution on an interval studied by several authors in recent past, and for which the the exponential rates of convergence to stationarity were computed analytically, but had no intuitive or probabilistic interpretation, except for the case where the jumps from the endpoints are identical (or more generally have the same distribution). We study convergence to stationarity probabilistically, by finding an efficient coupling. The coupling identifies the "bottlenecks" responsible for the rates of convergence and also gives tight computable bounds on the total variation norm of the process between two starting points. The adaptation to the diffusion case is straightforward.

math.PR↗

arXiv2012

Random walks on barycentric subdivisions and the Strichartz hexacarpet

We investigate the relation between simple random walks on repeated barycentric subdivisions of a triangle and a self-similar fractal, Strichartz hexacarpet, which we introduce. We explore a graph approximation to the hexacarpet in order to establish a graph isomorphism between the hexacarpet approximations and Barycentric subdivisions of the triangle, and discuss various numerical calculations performed on the these graphs. We prove that equilateral barycentric subdivisions converge to a self-similar geodesic metric space of dimension log(6)/log(2), or about 2.58. Our numerical experiments give evidence to a conjecture that the simple random walks on the equilateral barycentric subdivisions converge to a continuous diffusion process on the Strichartz hexacarpet corresponding to a different spectral dimension (estimated numerically to be about 1.74).

math.MG↗