arXiv subjects

Kesav Krishnan

Publications and source records attributed to Kesav Krishnan.

6 recordsLinked to original sources

arXiv2026

CLT for the Weight of the Minimum Spanning Arborescence on the Complete Directed Graph

The minimum spanning arborescence (MSA) with a fixed boundary, defined on weighted directed graphs, is the directed analogue of the minimum spanning tree (MST). The behaviour of the MSA has been of recent interest and is known to be quite different from that of its undirected counterpart. On the complete directed graph, and for edge weights given by i.i.d. random variables with a broad class of distributions, we prove a Central Limit Theorem for the weight of the MSA with appropriate scaling. While the scaling for the CLT is similar to MST case explored by Janson, the centering required ($1$ as opposed to $ζ(3)$ for Uniform$[0,1]$ weights) and the limiting variance of the Gaussian ($1$ as opposed to $6ζ(4)-4ζ(3)$ for Uniform$[0,1]$ weights) are notably different.

math.PR↗

arXiv2025

Local limit of the focusing discrete NLS

We examine the behavior of a function sampled from the invariant measure associated to the focusing discrete Non Linear Schrödinger equation, defined on a discrete torus of dimension $d \geq 3$, and nonlinearity parameter $p>4$, in the infinite volume limit. The Gibbs measure has two parameters, the inverse temperature and the strength of the non linearity, and the scaling is such that the non linear and linear parts of the Hamiltonian contribute on the same scale. It was shown by Dey, Kirkpatrick and the first author that this measure undergoes a phase transition, concerning concentration of mass of a typical function {at the level of partition functions}. We prove that the three regions of the phase diagram yield three distinct local limits, a massive Gaussian free field, a massless Gaussian free field plus a random constant, and finally a (possibly trivial) mixture of massive Gaussian free fields, where some mass is ``lost'' to the region of concentration. Our proof relies on the analysis of the spherical model of a ferromagnet. The measure under consideration is an exponential tilt of the spherical model, and the additional tilt can be seen to induce an additional phase transition. Although our motivations come from the NLS equation and soliton resolution conjecture, our proofs are completely probabilistic, and can be read without any knowledge of PDE theory.

math.PR↗

arXiv2024

On the local convergence of integer-valued Lipschitz functions on regular trees

We study random integer-valued Lipschitz functions on regular trees. It was shown by Peled, Samotij and Yehudayoff that such functions are localized, however, finer questions about the structure of Gibbs measures remain unanswered. Our main result is that the weak limit of a uniformly chosen 1-Lipschitz function with 0 boundary condition on a $d$-ary tree of height $n$ exists as $n \to \infty$ if $2 \le d \le 7$, but not if $d \ge 8$, thereby partially answering a question posed by Peled, Samotij and Yehudayoff. For large $d$, the value at the root alternates between being almost entirely concentrated on 0 for even $n$ and being roughly uniform on $\{-1,0,1\}$ for odd $n$, leading to different limits as $n$ approaches infinity along evens or odds. For $d \ge 8$, the essence of this phenomenon is preserved, which obstructs the convergence. For $d \le 7$, this phenomenon ceases to exist, and the law of the value at the root loses its connection with the parity of $n$. Along the way, we also obtain an alternative proof of localization. The key idea is a fixed point convergence result for a related operator on $\ell^\infty$, and a procedure to show that the iterations get into a `basin of attraction' of the fixed point. We also prove some accompanying analogous `even-odd phenomenon' type results about $M$-lipschitz functions on general non-amenable graphs with high enough expansion (this includes for example the large $d$ case for regular trees). We also prove a convergence result for 1-Lipschitz functions with $\{0,1\}$ boundary condition. This last result relies on an absolute value FKG for uniform 1-Lipschitz functions when shifted by $1/2$.

math.PR↗

arXiv2024

Uniqueness and CLT for the Ground State of the Disordered Monomer-Dimer Model on $\mathbb{Z}^{d}$

We prove that the disordered monomer-dimer model does not admit infinite volume incongruent ground states in $\mathbb{Z}^d$ which can be obtained as a limit of finite volume ground states. Furthermore, we also prove that these ground states are stable under perturbation of the weights in a precise sense. As an application, we obtain a CLT for the ground state weight for a growing sequence of tori. Our motivation stems from a similar and long standing open question for the short range Edwards-Anderson spin glass model.

math-ph↗

arXiv2023

Phase Transition for Discrete Non Linear Schrödinger Equation in Three and Higher Dimensions

We analyze the thermodynamics of the focusing discrete nonlinear Schrödinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$ and under a model with two parameters, representing inverse temperature and strength of the nonlinearity, respectively. We prove the existence of limiting free energy and analyze the phase diagram for general $d,p$. We also prove the existence of a continuous phase transition curve that divides the parametric plane into two regions involving the appearance or non-appearance of solitons. Appropriate upper and lower bounds for the curve are constructed. We also look at the typical behavior of a function chosen from the Gibbs measure for certain parts of the phase diagram.

math.AP↗

arXiv2021

Disordered Monomer-Dimer model on Cylinder graphs

We consider the disordered monomer-dimer model on cylinder graphs $\mathcal{G}_n$, i.e., graphs given by the Cartesian product of the line graph on $n$ vertices, and a deterministic graph. The edges carry i.i.d. random weights, and the vertices also carry i.i.d. random weights, not necessarily from the same distribution. Given the random weights, we define a Gibbs measure on the space of monomer-dimer configurations on $\mathcal{G}_n$. We show that the associated free energy converges to a limit, and with suitable scaling and centering, satisfies a central limit theorem. We also show that the number of monomers in a typical configuration satisfies a law of large numbers and a central limit theorem with appropriate centering and scaling. Finally, for an appropriate height function associated with a matching, we show convergence to a limiting function and prove the Brownian motion limit about the limiting height function in the sense of finite-dimensional distributions.

math.PR↗