Hi, I am Mriganka! I am a 4th year Doctoral Candidate at the Department of Statistics, UC
Berkeley, supervised by Shirshendu
Ganguly. I am broadly interested in various topics in probability, including
*spin systems, spectra of matrices, random graphs, Gibbs measures,* and *mixing times of Markov chains*. More recently,
I have also been exploring various directions in the theory of *neural networks* and *machine learning*.

*Feel free to click through some of the sections below. Or take a look at my (nascent)
blog.*

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 \(L = R = -\infty.\)

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).

- Spring 24, STAT 205B (Probability Theory part 2) — Prof. Shirshendu Ganguly
- Spring 24, STAT 206 (Advanced Topics in Probability and Stochastic Processes) — Prof. Steve Evans
- Fall 24, STAT 154/254 (Modern Statistical Prediction and Machine Learning) — Prof. Nikita Zhivotovskiy

Prior to that, I have been a GSI for STAT 205A (Graduate Probability Theory Part 1), 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).

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.

This website is created using flask (and by implication, python), with
tailwind for styling. The majority of the content here is rendered from markdown, and
structural information (for instance, the data for the papers or the seminar) is recorded as `.toml`

files, which is
parsed and loaded into jinja templates. A small bit of vanilla Javascript
is used to manage the toggles and render the \(\mathit{math}\). Some
localStorage is used to store data in your
browser to remember your choice of dark mode and toggled entries.

An older version of this website was written using rust and actix. A significant amount of
boilerplate was invested in type-checking `.toml`

files, something I have since found useless since all the parsing
happens at server start and hence errors can be caught at that point (surprisingly, actix itself didn't need much code,
and I was pleased with the UX). As a result, the 500-line codebase is now just less than 90 lines of python
(discounting the blog feature which was not present earlier).