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).