RESEARCH

Research Interests

I work broadly in the field of probability theory. In particular, I am mostly excited by problems originating in statistical mechanics and mathematical physics. I also dabble in combinatorics, ergodic theory and theoretical statistics from time to time. 

Most recently, I have been thinking about various random surface models which conjecturally have a conformally invariant scaling limits (which in most cases is the Gaussian free field). I am also interested in understanding the behaviour of various types of gradient models with hard core constraints.

In another direction, I am also interested in understanding whether a statistical mechanics model can be realized as a factor of iid and its various relations with the underlying geometry of the graph and the phase transition of the model.

In the past, I have worked in and around various aspects of random planar maps, Liouville quantum gravity and unimodular random graphs and uniform spanning trees. I also go back to these topics from time to time. 

Selected Publications

1.  Classification of half planar maps (with Omer Angel) In Ann. Probab. Volume 43, Number 3 (2015), 1315-1349.

2. Unimodular hyperbolic triangulations: circle packing and random walk (with Omer Angel, Tom Hutchcroft and Asaf Nachmias). In Inventiones Mathematicae.  206, 229-268, 2016.

3. Hyperbolic and parabolic unimodular random maps (with Omer Angel, Tom Hutchcroft and Asaf Nachmias). GAFA, 28, 879-942, 2018.

4. A short proof of the discontinuity of phase transition in the planar random-cluster model with q>4 (with Yinon Spinka). Comm. Math. Physics. 2020.  Talk by Yinon Spinka at BIRS.

5. Dimers and Imaginary geometry. (with Nathanael Berestycki and Benoit Laslier).  In Ann. Probab. 2020. A survey talk  about the project for general audience given at QM^3 seminar.  A survey talk by Nathanael Berestycki at JIPS.

6. Logarithmic variance for the height function of square-ice (With Hugo Duminil-Copin, Matan Harel, Benoit Laslier and Aran Raoufi). In Commun. Math. Phys. 2022.

Full list of publication and preprints

Gaussian free field and related surface  models

• (1+epsilon)-moments suffice to characterise the GFF  (with Nathanael Berestycki and Ellen Powell). In EJP. 2020 

• A short proof of the discontinuity of phase transition in the planar random-cluster model with q>4 (with Yinon Spinka). In Commun. Math. Phys. 2020. 

• A characterisation of the Gaussian free field (with Nathanael Berestycki and Ellen Powell). in PTRF, 2019.

• Logarithmic variance for the height function of square-ice (With Hugo Duminil-Copin, Matan Harel, Benoit Laslier and Aran Raoufi). In Commun. Math. Phys. 2022.

Dimer Model and scaling limits

• The dimer model on Riemann surfaces, I : Temperleyan forests (with Nathanael Berestycki and Benoit Laslier), AIHPD, Combinatorics, Physics and their Interactions. To appear.

• The dimer model on Riemann surfaces, II: conformal invariance and scaling limit. (with Nathanael Berestycki and Benoit Laslier), preprint.

• Dimers and Imaginary geometry. (with Nathanael Berestycki and Benoit Laslier).  Ann. Probab. Vol 48. No 1, 1-52, 2020.

• A note on dimers and T graphs (with Nathanael Berestycki and Benoit Laslier). Preprint. 

• Quantitative Russo-Seymour-Welsh for random walk on random graphs and decorrelation of UST (with Tingzhou Yu). In J. Theoret. Prob. 2022

Statistical physics models as Factors of i.i.d.

• Characterizations of amenability through stochastic domination and factors of i.i.d. (with Yinon Spinka). Preprint.

• Uniform even subgraphs and graphical representations of Ising as factors of i.i.d. (with Omer Angel and Yinon Spinka). In EJP. To appear.

• Proper planar 3-colorings are Bernoulli. (joint with Yinon Spinka). In Ergod. Theory Dyn. Syst. 2021.

• Finitary codings for gradient models and a new graphical representation for the six-vertex model (With Yinon Spinka). In Random Structures and Algorithms. 2021.

• A tale of two balloons (with Omer Angel and Yinon Spinka). In PTRF. 2022

Random planar maps, trees, and unimodular random graphs.

• Minimal Spanning Arborescence (with Arnab Sen). Preprint.

• Forests on wired trees (with Ben Xiao). In ALEA. 2022.

• A unimodular hyperbolic souvlaki- an appendix to "A liouville hyperbolic Souvlaki" (with Gabor Pete). In EJP. 2017

• Hyperbolic and parabolic unimodular random maps (with Omer Angel, Tom Hutchcroft and Asaf Nachmias). GAFA, 206, 229-268, 2018.

• Classification of scaling limits of uniform infinite planar quadrangulations with a boundary. (with Erich Baur and Gregory Miermont). Ann. Probab.  2019, Vol. 47, No. 6, 3397-3477.

• The half plane UIPT is recurrent.  (with Omer Angel). In PTRF

• Critical exponents on Fortuin-Kasteleyn weighted planar maps (with Nathanael Berestycki and Benoit Laslier). In Commun. Math. Phys., 2017.

• Unimodular hyperbolic triangulations: circle packing and random walk (with Omer Angel, Tom Hutchcroft and Asaf Nachmias).  In Inventiones Mathematicae. 2015.

• Random walks on stochastic hyperbolic half planar triangulations (with Omer Angel and Asaf Nachmias). In Random Structures and Algorithms. 2014

• Geometry and percolation on half planar triangulations. In Elec. J. Prob. 2014. 

• Large unicellular maps in high genus. In AIHP. 2014.

• The local limit of unicellular maps in high genus. (with Omer Angel, Guillaume Chapuy and Nicolas Curien). In Elec. Comm. Prob. 2013

• Classification of half planar maps (with Omer Angel) In Ann. Probab. 2013. 

Signal Detection problem in Statistical physics models

• On Testing for Parameters in Ising Models. (with Rajarshi Mukherjee). In AIHP. 2020

• Sharp signal detection in ferromagnetic Ising models. (with Rajarshi Mukherjee and Sohom Bhattacharya). Preprint.

Slides and videos from some talks

• Mini course at ICTS on Dimers and Imaginary geometry. Here is a talk with an overview of the topic. 

• Percolation today talk on Loop O(1) as factor of iid. Jointly with Yinon Spinka. Talk by Omer Angel on the same topic at FPSAC 2022.

• Talk at Northwest probability seminar on characterizing the Gaussian free field. Some slides on the same topic.

• Logarithmic variance of square ice (some of the slides here are due to Matan Harel).

• A short proof of the discontinuity of phase transition in the planar random-cluster model with q>4  (slides due to Yinon Spinka)

Students

• Tingzhou Yu (Masters candidate). 2019-21

• Ben Xiao (Masters Candidate). 2020-22

• Yakov Shklarov (Masters Candidate),  joint with A. Quas. 2021-23

• Nathaniel Butler (Masters Candidate). 2022-

• Swarnadeep Bagchi (PhD Candidate). 2023-

Postdocs

• Kesav Krishnan (2023-)

Coauthors

I have had the privilege to work and write papers with Omer Angel, Erich Baur, Nathanael Berestycki, Sohom Bhattacharya, Nicolas Curien, Guillaume Chapuy, Hugo Duminil-Copin, Matan Harel, Tom Hutchcroft, Benoit Laslier, Gregory Miermont, Krishanu Maulik, Rajarshi Mukherjee, Asaf Nachmias, Gabor Pete, Ellen Powell, Aran Raoufi, Yinon Spinka.