# Current Research Interest

# Introduction

My research interest is in integrable combinatorics and probability,
which arises from the intersection of combinatorics, statistical
mechanics, representation theory and mathematical physics. I am particularly interested in problems revolves around the \(A_\infty\) \(T\)-system in the mathematical physics
literatures, also known as octahedron recurrence relations when associated to \(A_\infty\) Lie algebra. Some of the research directions are reviewed on this page

# The \(T\)-system

The \(A_\infty\) \(T\)-system or octahedron relation is the
following recursion relation for variables \(T_{i,j,k}>0\), \(i,j,k\in {\mathbb Z}\) \[\label{Tsys}
T_{i,j,k+1}\, T_{i,j,k-1}=T_{i+1,j,k}\, T_{i-1,j,k}+T_{i,j+1,k}\,
T_{i,j-1,k} .\] It may be interpreted as a discrete time \(k\) evolution for the variable \(T\), expressing its value at the time \(k+1\) vertex of an octahedron in terms of
the values at the 4 vertices at time \(k\) and at a single vertex at time \(k-1\). The solution \(T_{i,j,k}\) is unique once we fix
admissible initial data along any given “stepped surface" \({\mathbf k}\) made of the vertices \((i_0,j_0,k_{i_0,j_0})\), \(i_0,j_0\in {\mathbb Z}\), where the height
function \(k_{i,j}:{\mathbb Z}^2\to{\mathbb
Z}\) obeys \(|k_{i+1,j}-k_{i,j}|=|k_{i,j+1}-k_{i,j}|=1\)
for all \(i,j\in {\mathbb Z}\). My
research focuses on a particular solution of the \(A_\infty\) \(T\)-system subject to initial data along
\((r,s,t)\)-slanted parallel planes
\[(P_m)=\{(i,j,k)\, \vert \, r i+s j+t
k=m\}\] for some fixed integers \(r,s,t\geq 0\) such that \(t>\max(r,s)\) and \(\gcd(r,s,t)=1\). Together with Di
Francesco, we prove the following theorem extending the previous work
in^{1}
to a specific class of graph known as *pinecone* in^{2}:

**Theorem 1**. ^{3} The
solution of the \(T\)-system with
slanted initial data is expressed as: \[T_{i,j,k}=\sum_{{\rm dimer}\, {\rm configs.}\,
D\atop
{\rm on}\, \mathcal G} \prod_{{\rm faces}\, (x,y)\atop {\rm
of}\, G}\begin{cases}
(t_{x,y})^{v_{x,y}/2-1-N_{x,y}(D)} & (x,y)
\ \text{interior faces} \\
(t_{x,y})^{1-N_{x,y}(D)} & (x,y) \ \text{boundary
faces}
\end{cases}\] where the sum extends over all dimer
configurations \(D\) on the pinecone
graph \(\mathcal G\), while \(v_{x,y}\) is the valency of the face \((x,y)\) and \(N_{x,y}(D)\in \{0,1,...,v_{x,y}\}\) denotes
the number of dimers occupying the edges at the boundary of the face
\((x,y)\).

# ACSV and Arctic Curves

For fixed values of \((r,s,t)\) the simplest solution of the \(T\)-system [Tsys] corresponds to choosing uniform initial data in each initial data plane \((P_\ell)\), \(\ell=0,1,...,2t-1\). More precisely, choosing the initial values of \(T\) to be \(T_{i,j,k}=a_\ell\) for all \((i,j,k)\in (P_\ell)\) for some positive real numbers \(a_0,a_1,...,a_{2t-1}\), we deduce that for all \(m\geq 2t\): \[T_{i,j,k}=a_m \qquad (i,j,k)\in (P_m)\] where \(a_m\), \(m\geq 2t\) are subject to the “Gale-Robinson" recursion relation \[a_m \, a_{m-2t}=a_{m+r-t}\, a_{m-r-t}+a_{m+s-t}\, a_{m-s-t}\] Among these solutions a particularly simple one consists in taking \(a_\ell={\alpha}^{\ell(\ell-1)/2}\) for \(\ell=0,1,...,2t-1\), leading to \(\displaystyle a_m={\alpha}^{m(m-1)/2},(m\in {\mathbb Z}),\) provided \({\alpha}\) satisfies \(\displaystyle{\alpha}^{t^2} ={\alpha}^{r^2}+{\alpha}^{s^2}\).

Pick a point \((i_0,j_0,k_0=k_{i_0,j_0})\) belonging to
one of the initial data planes \((P_{ri_0+sj_0+tk_0})\) with \(0\leq r i_0+s j_0+t k_0<2t\)). Assume it
corresponds in the dimer graph to the center of a \(2v\)-valent face. As the local contribution
for this face to the partition function is \((t_{i_0,j_0})^{v-1-N_{i_0,j_0}(\mathcal
D)}\), we may write \[\label{averageD}
\rho^{(i_0,j_0,k_0)}_{i,j,k}:=\frac{1}{T_{i,j,k}}\, t_{i_0,j_0}
\partial_{t_{i_0,j_0}} (T_{i,j,k}) =\langle v-1- N_{i_0,j_0}(\mathcal D)
\rangle_{i,j,k}\] where \(\langle f
\rangle_{i,j,k}\) stands for the statistical average of the
function \(f\) over the dimer
configurations \(\mathcal D\) for the
\((i,j,k)\) dimer model, and where
\(k_0=k_{i_0,j_0}\) indicates the time
variable along the initial data surface. This density is an
*order parameter* for the crystalline/liquid phases
of the model, namely \(\rho^{(i_0,j_0,k_0)}_{i,j,k}\) vanishes
identically in the *crystal phase*, while it fluctuates and
becomes non-zero in the *liquid* regions. Via the framework of
Analytic Combinatorics in Several Variables (ACSV) developed by
Baryshnikov, Pemantle and Wilson^{4–6}, we
computed the arctic curves of the octahedron recurrence with some
special initial conditions:

**Theorem 2**. ^{3} The limit
shape of typical large size (r,s,t)-pinecone domino tilings associated
to the solution of the \(T\)-system
with uniform initial data \(t_{i,j}={\alpha}^{m(m-1)/2}\) on each
slanted plane \(m=ri+sj+tk = 0,1, \cdots,
2t-1\), is the ellipse \[\label{arcticu}
(1-A)\,t^2u^2+A\,t^2v^2-A(1-A)\,(r u+s v+t)^2 =0\] where \(A=A_{r,s,t}:={\alpha}^{r^2-t^2},\qquad
1-A={\alpha}^{s^2-t^2}\) inscribed in the scaling domain \[\begin{aligned}
v&=-\frac{t}{t+s}-\frac{t+r}{t+s}\, u , \qquad
v=-\frac{t}{s-t}-\frac{t+r}{s-t}\, u
\qquad \left(u\in\left[-\frac{t}{t+r},0\right]\right)\\
v&=-\frac{t}{t+s}+\frac{t-r}{t+s}\, u, \qquad
v=-\frac{t}{s-t}+\frac{t-r}{s-t}\, u \qquad \left(u\in
\left[0,\frac{t}{t-r}\right]\right) .
\end{aligned}\]

By introducing the periodic initial data, we computed more arctic curves of this dimer model with now additional gaseous phases in Fig. 1

,

# Newton Polygon and Integrable System

Kenyon, Okounkov and Sheffield developed the theories of limit shapes
for dimer on embedded bipartite periodic graphs in^{7,8}. Without considering
the weight from the previous section, the graph coming from the set of
parallel planes can be realized as a periodic graph on a torus:

**Example 3**. *The slanted \((1,1,3)\) \(T\)-system initial data viewed as graph on
torus is in Fig. 2*

From the description of the Newton polygon, if the weights on this bipartite graph on the torus is fully periodic, then one should witness five frozen regions. Our result from the previous section, however provides in the scaling limits with only four frozen regions in the uniform case and with gaseous regions in the case of periodic weights on each initial data planes. A natural question to ask from this result would be: Is there a gauge equivalent transformation in weights mapping the limit curves coming from two different methods. This question will be discussed in an upcoming research.

It is also worth mentioning that the zigzag paths, with edge weights
on the bipartite dimer graph can be used to compute the Hamiltonians and
Casimirs of the cluster integrable system introduced by^{9}.
In^{10},
Vichitkunakorn proved that the Hamiltonians coming from the cluster
integrable system is consistent with the conserved quantities of the
\(Q\)-system, which can be seen as the
*combinatorial limit* of the \(T\)-system. One then can ask the question
of finding the relations between the conserved quantities of the \(T\)-system (some of these are mentioned
in^{11–13}) and the
Hamiltonians of the cluster integrable system coming from the dimer
model corresponding to \(T\)-system
initial data.

# \(t\)-embedding

Recently, another interesting research in dimer model lies in the
work of *\(t\)-embedding* (also
known as *Culomb gauges*) by^{14,15} and the
application of the discrete complex analysis techniques in probability
and statistical physics. The existence of *perfect \(t\)-embedding* is still an open
question as the construction of the object is susceptible to the
boundary conditions. In 2023, Bergren, Russkikh and Nicoletti showed the
existence of perfect \(t\)-embedding
for a class of graph called *tower graph*, and previously Chelkak
and Ramassamy^{16} showed the
existence for the well-celebrated Aztec Diamond. Both of this graph can
be equivalently defined for a suitable \((r,s,t)\)-slanted initial data from the
\(T\)-system. In future research, we
want to show the existence of (perfect) \(t\)-embedding for this class of bipartite
graph from the \(T\)-system initial
data.

**Conjecture 4**. *There exists a \(t\)-embedding, potentially proper and
perfect \(t\)-embedding, for the
pinecone graph of the \((r,s,t)\)-slanted initial data*

Once the existence of \(t\)-embedding is proved, one can use the
notion of discrete holomorphicity to analyze the scaling limit and prove
fluctuation results of the dimer model’s *height function* to
*standard Gaussian free field*^{14}.

# \(q\)-TASEP and Difference Operator

We mentioned the "combinatorial limit" of the \(T\)-system under the name \(Q\)-system in the previous section.

**Definition 5**. *Let \({\mathfrak{g}}\) be a simple Lie algebra
with Cartan matrix \(C\), with a set of
simple root index \(I_r=\{1, \ldots,
r\}\). The \(Q\)-system is the
family of variables \(\{Q_{{\alpha},k}| ~
{\alpha}\in I_r, k\in {\mathbb Z}\}\) satisfying: \[\label{Qsystem}
Q_{{\alpha},k+1} Q_{{\alpha},k-1} = Q_{{\alpha},k}^2 -
\prod_{\beta\sim{\alpha}}
\mathcal T^{({\alpha},\beta)}_k, ~{\alpha}\in I_r, ~k\in {\mathbb
Z},\] where \({\alpha}\sim\beta\) means \({\alpha}\) is connected to \(\beta\) in the Dynkin diagram of \({\mathfrak{g}}\), and \[\label{T}
\mathcal T^{({\alpha},\beta)}_k = \prod_{i=0}^{|C_{{\alpha},\beta}|-1}
Q_{\beta,\lfloor\frac{t_\beta k+i}{t_{\alpha}} \rfloor},\] where
\(\lfloor a \rfloor\) is the integer
part of \(a\). Here \(t_{\alpha}\) are the integers symmetrizing
the Cartan matrix. Namely, \(t_r=2\)
for \(B_r\), \(t_{\alpha}=2\) (\({\alpha}<r\)) for \(C_r\), \(t_3=t_4=2\) for \(F_4\) and \(t_2=3\) for \(G_2\), and \(t_{\alpha}=1\) otherwise.*

This system admits the structure of cluster algebra and
integrability^{10,17,18} and
rich combinatorial properties^{19,20}.

Switching to a different point of view, Corwin, Borodin and Sasamoto
studied the \(q\)-deformed Totally
Asymmetric Exclusion Process (\(q\)-TASEP)^{21–23}.
The \(q\)-moments, i.e. the family of
expectations \(\operatorname{\mathbb{E}}\left[q^{k(X_N(\tau)+N)}
\right]\), where \(k=0,1,2,\ldots\) can be computed via the
*Macdonald measure* at \(t = 0\)
by applying the Macdonald operator \(D_N^1\) to the Cauchy identity and
exploiting the Markov duality with another process^{23}. On the
other hand, the Macdonald operators are known to satisfied the "quantum"
version of the \(Q\)-system in a series
of work by Kedem and Di Francesco^{24–27}.
A future research problem that was proposed by Leonid Petrov during the
research program in Geometry, Statistical Mechanics, and Integrability
at IPAM is to consider two different \(q\)-TASEPs:

\(X_1,X_2,\ldots\) with speed parameters \(x_1,x_2,\ldots,x_{N-1},x_N,x_{N+1} ,\ldots\)

\(\tilde X_1,\tilde X_2,\ldots\) with speed parameters \(x_1,x_2,\ldots,x_{N-1},x_{N+1},x_{N},\ldots\), which differ from the first one by the swap of \(x_{N}\) and \(x_{N+1}\).

Let both processes start from the step initial configuration \(X_i(0)=-i\), \(\tilde X_i(0)=-i\). Denote by \(D_N\) and \(\tilde D_N\) the first Macdonald operators \(D_N^1\) acting in the variables \(x_1,\ldots,x_N\) and \(x_1,\ldots,x_{N-1},x_{N+1}\), respectively, and specialized at \(t=0\). In some unpublished work, Petrov and Ahn wrote down the following relation \[(\tilde D)^k= \sum_{j=0}^k \varphi_{q,x_{N+1}/x_N}(j\mid k)\hspace{1pt} (D_N)^j(D_{N+1})^{k-j}, \qquad \varphi_{q,\alpha}(j\mid k):= \alpha^j(\alpha;q)_{k-j}\frac{(q;q)_k}{(q;q)_j(q;q)_{k-j}}.\] When \(0\le \alpha\le 1\), the quantities \(\varphi_{q,\alpha}(j\mid k)\) form a probability distribution in \(j\in \left\{ 0,1,\ldots,k \right\}\). It is worth to consider the \(t\)-deformation of the relation above. For example, when \(k=1\), we have \[\label{eq:relation_for_D_at_t_0} \tilde D = \left( 1-\frac{x_{N+1}}{x_N} \right)D_N + \frac{x_{N+1}}{x_N}D_{N+1}.\] with the \(t\)-deformation \[\label{t_analog_diff} \tilde D = \frac{x_{N+1}-x_N}{tx_{N+1}-x_N}\hspace{1pt}D_{N+1} + \frac{t x_{N}-x_{N+1}}{tx_{N+1}-x_N}\hspace{1pt}D_{N} + \frac{t(x_{N+1}-x_N)}{tx_{N+1}-x_N}\hspace{1pt}D_{N-1}.\] These \(t\)-deformed power relations may be applicable in the study of stochastic particle systems and random matrix models. For \(q=t\), the relations might be useful in the study of GUE corners process and related matrix models with unitary symmetry.

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**16**(1), Research Paper 125, 37 (2009).

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