About

Hi, I am Mriganka! I am a 3rd year Doctoral Candidate at the **Department of Statistics, UC Berkeley**, supervised by Shirshendu Ganguly. I am broadly interested in various topics in modern probability, including spin systems, spectra of matrices, random graphs, Gibbs measures, and mixing times. You may find my academic resume here and my industrial resume here (they are slightly different in emphasis and conciseness).

My student email address is:` `

`mriganka_brc at berkeley dot edu`

My personal email address is:` `

`mbrc12 at gmail dot com`

Papers

For each paper below, you will find an associated description.
It is intended to be an accessible explanation of the main goal of the work.
Detailed abstracts will be found in the link to the paper.

Characterizing Gibbs states for area-tilted Brownian lines

2023,
joint work with Pietro Caputo and Shirshendu Ganguly.
[paper]
[slides]
▸ Show description

Models in statistical physics often give rise to "line ensembles". For example, the classical Solid-on-Solid model seeks to study the interface between two solids. More specifically, it studies the interface (between the two spins) in a magnet whose, say, bottom surface is magnetized to be spin "down", and rest of the magnet is spin "up" (this is not representative of the actual physics of the system, the true interpretation and model comes from quantum mechanical considerations). Many important quantities of interest may then be examined by understanding the "level curves" of this object -- which are curves on this surface passing through areas which are at a given height from the equilibrium height of the interface. The local behavior of these curves is of great interest, and suitable probabilistic objects have been defined which capture that behavior. The *area-tilted line ensemble* arises out of these considerations, and is an infinite family of random curves (called "lines"), ordered one below the other, all above the x-axis. Think of each such line as one level curve, scaled appropriately.

To elaborate further, this ensemble is specified in terms of its "finite domain marginals". The so-called "area-tilted Brownian Gibbs" property prescribes this behavior and is a consequence of the Solid-on-Solid origins of this model. It is known that this ensemble is a well-defined mathematical object (nothing blows up or vanishes under the scaling operations performed to reach it). Until now, only one ensemble was known which satisfied the said Gibbs property. In this paper, we succeed in finding *all of them*. Perhaps surprisingly, there is an infinite family of such ensembles (yes, an infinite family of families of infinitely many lines) parametrized by two numbers $L, R$ (representing left and right "rates of growth") such that their sum is $L + R < 0$. The original ensemble corresponds to $L = R = -\infty$.

Universality in prelimiting tail behavior for regular subgraph counts in the Poisson regime

▸ Show description

A graph is a mathematical object used to describe relationships between things. Since real-world graphs are very complex, it is often of interest to understand the nature of *random* graphs. The most famous model of random graphs is the celebrated Erdős–Rényi model, which is created by taking $n$ vertices, and joining each pair with probability $p$ independently - let $G$ be such a random graph. Think of this as $n$ people, and any two people know each other with probability $p$ independently of the others. For $p = 0.1$ say, each vertex is connected to $0.1 n$ many other vertices on average, which is a huge number. Many applications require this average *degree*, i.e., the number of other vertices a given vertex is connected to, to be constant, say 5. A moment's thought then reveals that one may seek to emulate this behavior by *modifying $p$ with $n$*, precisely by choosing something like $p = 5/n$. This is a so-called *sparse* Erdős–Rényi graph.

Several basic questions can be asked about this model. Among them, of enormous importance is the notion of *subgraph count*. For example, I can ask "How many triangles are there in this graph?" by which I mean "How many groups of 3 people A, B, C are there such that all of them know each other?". Our understanding of how exactly this count behaves (on a very precise level) was actually open until very recently. But as one may imagine, it is possible to ask this question for *any configuration*, say the count of *four people* A, B, C, D forming a "square". In this spirit, this paper provides a generalization of the recent triangle result, extending it to *every possible "regular" subgraph*, which is every configuration where each person knows the same number of people among the rest (the square example is one such - everyone knows 2 others in the group).

Teaching

I am currently a GSI for STAT 154/254 (Modern Statistical Prediction and Machine Learning), which is being taught by Nikita Zhivotovskiy.

In the past, I have been a GSI for STAT 205A (which is a graduate-level probability class), STAT 134 (which is an upper-division probability class for undergraduates), and STAT 88 (which is a more introductory variant of STAT 134 combined with some emphasis on statistics).

Extras

Apart from research, I spend a significant amount of time coding, making simulations (often probabilistic in nature!) and (simple/incomplete) games. Although I am no longer active in the scene, I used to participate quite frequently in competitive programming competitions, for example, on Codeforces (mbrc) and Codechef (mbrc). Once a year, I would also participate in the now-defunct but celebrated Google Code Jam. In 2021, I (with my team of 3) qualified for the ICPC World Finals, held in Moscow, Russia.

These more "scientific" pursuits aside, I also enjoy gaming, listening to music, reading manga, and watching anime/TV shows. I participated (as a team of one) in one of the biggest game jams, the GMTK Game Jam 2023 - a 48-hour game jam based around a theme. My entry, *A Conscious Virus*, can be found here.