Organizing Committee

Limits of discrete random structures appear in different areas of probability, combinatorics, and machine learning. In statistical mechanics, probabilistic and combinatorial techniques are applied to rigorously describe the scaling limits of such random graphical models, which are closely related to phase transitions. In the vicinity of a phase transition, even a tiny change in some local parameter can result in dramatic changes in the macroscopic properties of the entire system. Random discrete structures are also useful mathematical models of large networks, which play a central role in our social and economic lives as the fabric over which we interact, form social connections, conduct economic transactions, transmit information, propagate disease, and much more.

The goal of this workshop is to integrate the algebraic combinatorics, probability, and machine learning paradigms of statistical mechanical models and to bring together researchers in related fields to discuss recent progress and new ideas.

Image for "Asymptotic Limits of Discrete Random Structures"

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Friday, September 29, 2023
  • 8:30 - 8:50 am EDT
    Check In
    11th Floor Collaborative Space
  • 8:50 - 9:00 am EDT
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 9:00 - 9:45 am EDT
    Formulas for Macdonald polynomials via the multispecies exclusion and zero range processes
    11th Floor Lecture Hall
    • Olya Mandelshtam, University of Waterloo
    We describe some recently discovered connections between one-dimensional interacting particle models and Macdonald polynomials and show the combinatorial objects that make this connection explicit. The first such model is the multispecies asymmetric simple exclusion process (ASEP) on a ring, linked to the symmetric Macdonald polynomials P_\lambda through its partition function, with multiline queues as the corresponding combinatorial object. The second particle model is the multispecies totally asymmetric zero range process (TAZRP) on a ring, which was recently found to have an analogous connection to the modified Macdonald polynomial H_\lambda. The combinatorial objects interpolating between probabilities of the TAZRP and the modified Macdonald polynomials turn out to be tableaux with a queue inversion statistic. We explain the plethystic relationship between multiline queues and queue inversion tableaux, and along the way, derive a new formula for P_\lambda using the queue inversion statistic. This plethystic correspondence is closely related to fusion in the setting of integrable systems.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Dimers in 3D
    11th Floor Lecture Hall
    • Catherine Wolfram, MIT
    A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. Given a compact region R in R^d, consider regions R_n in (1/n) Z^d which approximate R (and some boundary condition). What do random dimer tilings of R_n look like for large n? The 2D version of this question was answered by Cohn, Kenyon, and Propp in 2000. I will talk about how to answer this question in 3D. In both cases, it turns out that a random tiling of R_n is exponentially more likely to lie close to a fixed ""limit shape"" (and more generally, there is a large deviation principle, meaning the probability of lying close to any possible limiting configuration is given by a function called the ""rate function""). While the results are analogous, I will explain that the methods of proof are very different, because many of the key tools for studying dimers are special to two dimensions. This talk is based on, which is joint work with Nishant Chandgotia and Scott Sheffield.
  • 11:30 am - 12:15 pm EDT
    Dimer models and local systems
    11th Floor Lecture Hall
    • Haolin Shi, Yale University
    The dimer model studies a natural probability measure on the space of perfect matching (""dimer covers"") of an edge-weighted graph; the probability of a dimer cover is proportional to the product of its edge weights. When the graph is bipartite, a useful change in viewpoint is to view the edge weights as defining a ""C∗ local system"", that is a line bundle with connection. This leads us to consider natural generalizations using higher-rank bundles, in particular SLn -local systems. We will talk about a collection of results with emphasis on calculating connection probabilities in double and triple dimer models.
  • 12:25 - 12:30 pm EDT
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    Diagram algebras
    11th Floor Lecture Hall
    • Rosa Orellana, Dartmouth College
    One of the best-known planar diagram algebras is the Temperley-Lieb algebra. This algebra is defined combinatorially using non-intersecting matchings and can be realized as a centralizer of Lie algebras (or quantum groups) acting on tensor space. They have a wide range of applications, most notably to knot theory and statistical mechanics. The partition algebra is a generalization of the Temperley-Lieb algebra which was proposed by Martin to study higher dimensional statistical models. In this talk I will discuss planar subalgebras of the partition algebras related to the Temperley-Lieb algebra. This is joint work with N. Wallace and M. Zabrocki.
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EDT
    Restricted permutations
    11th Floor Lecture Hall
    • Richard Kenyon, Yale University
    We discuss large permutations with restricted permutation matrices, that is, whose permutation matrix has no 1s in some region. We give enumerative results and limit shapes.
  • 5:00 - 6:30 pm EDT
    11th Floor Collaborative Space
Saturday, September 30, 2023
  • 8:30 - 9:15 am EDT
    A Sanov-type theorem for Unimodular Marked Random Graphs, and its applications
    11th Floor Lecture Hall
    • Kavita Ramanan, Brown University
    We establish a large deviation principle in a strong topology for the component empirical measure of several sequences of marked random graph models, including Erdos-Renyi random graphs, random regular graphs, and more general configuration models. We show that the corresponding rate function is given by a relatively tractable formula involving the relative entropy functional. We also describe several applications of this result, such as Gibbs conditioning principles. This talk is based on joint work with I-Hsun Chen and Sarath Yasodharan.
  • 9:30 - 10:15 am EDT
    Permutation limits (Permutons)
    11th Floor Lecture Hall
    • Sumit Mukherjee, Columbia University
    Permutation limit theory arises by viewing a permutation as a probability measure on the unit square, and is motivated by dense graph limit theory. Using the theory of permutation limits (permutons), we can compute limiting properties of various permutation statistics for random permutations, such as number of fixed points, number of small cycles, pattern counts, and degree distribution of permutation graphs. We can also derive LDPs for random permutations. Our results apply to many non uniform distributions on permutations, including the the celebrated Mallows model, and mu-random permutations. This is based on joint work with Bhaswar Bhattacharya, Jacopo Borga, Sayan Das and Peter Winkler.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    Higher-Order Graphon and Permuton Theories: Fluctuations, Inference, and Degeneracies
    11th Floor Lecture Hall
    • Bhaswar Bhattacharya, University of Pennsylvania
    Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct confidence intervals for the motif densities. We will also present parallel results for patterns in random permutations through the lens of permuton theory. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution.
  • 12:00 - 1:30 pm EDT
    Lunch/Free Time
  • 1:30 - 2:15 pm EDT
    Interactive workshop on planning a new active learning probability course for research-level students in math-adjacent fields.
    11th Floor Lecture Hall
    • Robin Pemantle, University of Pennsylvania
  • 2:30 - 3:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:00 pm EDT
    Poster Session
    11th Floor Collaborative Space
  • 4:00 - 5:00 pm EDT
    Active Learning Process
    11th Floor Lecture Hall
    • Robin Pemantle, University of Pennsylvania
    The workshop will continue in an optional evening session, where we will make a plan for a course that fits the most populations of those attending the evening session. We will look over some existing materials, plan a shared materials archive, and discuss some of the more crucial mechanics of active classrooms.
Sunday, October 1, 2023
  • 9:00 - 9:45 am EDT
    Strong characterization of the Airy line ensemble
    11th Floor Lecture Hall
    • Jiaoyang Huang, University of Pennsylvania
    The Airy line ensemble was introduced by Prähofer and Spohn and is conjectured to describe the scaling limit of various random surfaces and stochastic growth models in the Kardar–Parisi–Zhang universality class. In this talk I will discuss a characterization result for Airy line ensembles, essentially indicating that if the top curve of a Brownian line ensemble is within a multiplicative 1+o(1) from a parabola, then it must be the Airy line ensemble (up to an affine shift). This is a joint work with Amol Aggarwal.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Structures in random graphs: New connections
    11th Floor Lecture Hall
    • Huy Tuan Pham, Stanford University
    The study of structures in large random graphs has been a central direction in probabilistic combinatorics. In this talk, I will survey several recent developments in this front with interesting connections. Certain important aspects in the study of structures in random graphs can be phrased in terms of thresholds — the density locations at which a structure emerges. In joint work with Jinyoung Park, building on connections to our resolution of a conjecture of Talagrand on the behavior of random linear programs under combinatorial constraints, we prove the long-standing Kahn-Kalai conjecture, that thresholds of general monotone properties are closely predicted by expectation obstructions. The Kahn-Kalai conjecture is a beautiful milestone towards the understanding of emergence of general structures, and yet to complete the quest, it remains to study these expectation obstructions. This latter task can prove to be highly challenging in several cases and bring in interesting connections to structural results. As an illustration, I will discuss joint work with Ashwin Sah, Mehtaab Sawhney and Michael Simkin on enumerating and determining the threshold of clique factors and bounded degree spanning trees in random subgraphs of graphs with high minimum degree. Our proof crucially builds on the regularity method and embedding techniques, which are seminal cornerstones of modern combinatorics. Switching from independent Erdos-Renyi random graphs to structured ensembles with dependency introduces significant challenges. I will discuss this challenge in the context of random Cayley subgraphs of general groups. Given a fixed finite group, random Cayley graphs are constructed by choosing the generating set at random. These graphs thus reflect interesting symmetries and properties of the group, at the cost of inducing complex dependencies. I will discuss results on clique and independence numbers in random Cayley graphs of general groups, as well as progress towards a conjecture of Alon on Cayley graphs with small clique and independence number. These questions are naturally connected with some fundamental problems in additive combinatorics, which we address using both group theoretic and purely combinatorial perspectives. This is based on joint work with David Conlon, Jacob Fox and Liana Yepremyan.
  • 11:30 am - 12:15 pm EDT
    Questions on Finite Graphs Motivated by Infinite Graphs
    11th Floor Lecture Hall
    • Virtual Speaker
    • Russell Lyons, Indiana University
    We discuss two questions for random walks on finite graphs that are motivated by an old conjecture of Benjamini, Schramm, and me and an old question of Fontes and Mathieu. These old questions concern random walks on infinite Cayley graphs, the first involving also percolation and the second involving random environments. We try to attack them via questions on finite graphs without any group structure.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).

All event times are listed in .

Application Information

ICERM welcomes applications from faculty, postdocs, graduate students, industry scientists, and other researchers who wish to participate. Some funding may be available for travel and lodging. Graduate students who apply must have their advisor submit a statement of support in order to be considered.

Your Visit to ICERM

ICERM Facilities
ICERM is located on the 10th & 11th floors of 121 South Main Street in Providence, Rhode Island. ICERM's business hours are 8:30am - 5:00pm during this event. See our facilities page for more info about ICERM and Brown's available facilities.
Traveling to ICERM
ICERM is located at Brown University in Providence, Rhode Island. Providence's T.F. Green Airport (15 minutes south) and Boston's Logan Airport (1 hour north) are the closest airports. Providence is also on Amtrak's Northeast Corridor. In-depth directions and transportation information are available on our travel page.
ICERM's special rate will soon be made available via this page for our preferred hotel, the Hampton Inn & Suites Providence Downtown. ICERM also regularly works with the Graduate Hotel and Hilton Garden Inn who both have discounted rates available. Contact before booking anything.
The only way ICERM participants should book a room is through the hotel reservation links located on this page or through links emailed to them from an ICERM email address ( ICERM never works with any conference booking vendors and never collects credit card information.
Those traveling with family who are interested in information about childcare and/or schools should contact
Technology Resources
Wireless internet access ("Brown-Guest") and wireless printing is available for all ICERM visitors. Eduroam is available for members of participating institutions. Thin clients in all offices and common areas provide open access to a web browser, SSH terminal, and printing capability. See our Technology Resources page for setup instructions and to learn about all available technology.
To request special services, accommodations, or assistance for this event, please contact as far in advance of the event as possible. Thank you.
Discrimination and Harassment Policy
ICERM is committed to creating a safe, professional, and welcoming environment that benefits from the diversity and experiences of all its participants. Brown University's "Code of Conduct", "Discrimination and Workplace Harassment Policy", "Sexual and Gender-based Misconduct Policy", and "Title IX Policy" apply to all ICERM participants and staff. Participants with concerns or requests for assistance on a discrimination or harassment issue should contact the ICERM Director; they are the responsible employees at ICERM under this policy.
Fundamental Research
ICERM research programs aim to promote Fundamental Research and mathematical sciences education. If you are engaged in sensitive or proprietary work, please be aware that ICERM programs often have participants from countries and entities subject to United States export control restrictions. Any discoveries of economically significant intellectual property supported by ICERM funding should be disclosed.
Exploring Providence
Providence's world-renowned culinary scene provides ample options for lunch and dinner. Neighborhoods near campus, including College Hill Historic District, have many local attractions. Check out the map on our Explore Providence page to see what's near ICERM.

Visa Information

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B-1 or Visa Waiver Business (WB)
Ineligible to be reimbursed
B-2 or Visa Waiver Tourist (WT)
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F-1 and J-1 not sponsored by ICERM: need to obtain a letter approving reimbursement from the International Office of your home institution PRIOR to travel.

H-1B holders do not need letter of approval.

All other visas: alert ICERM staff immediately about your situation.

ICERM does not reimburse visa fees. This chart is to inform visitors whether the visa they enter the US on allows them to receive reimbursement for the items outlined in their invitation letter.

Financial Support

This section is for general purposes only and does not indicate that all attendees receive funding. Please refer to your personalized invitation to review your offer.

As this program is funded by the National Science Foundation (NSF), ICERM is required to collect your ORCID iD if you are receiving funding to attend this program. Be sure to add your ORCID iD to your Cube profile as soon as possible to avoid delaying your reimbursement.
Acceptable Costs
  • 1 roundtrip between your home institute and ICERM
  • Flights in economy class to either Providence airport (PVD) or Boston airport (BOS)
  • Ground Transportation to and from airports and ICERM.
Unacceptable Costs
  • Flights on U.K. airlines
  • Seats in economy plus, business class, or first class
  • Change ticket fees of any kind
  • Multi-use bus passes
  • Meals or incidentals
Advance Approval Required
  • Personal car travel to ICERM from outside New England
  • Multiple-destination plane ticket; does not include layovers to reach ICERM
  • Arriving or departing from ICERM more than a day before or day after the program
  • Multiple trips to ICERM
  • Rental car to/from ICERM
  • Arriving or departing from airport other than PVD/BOS or home institution's local airport
  • 2 one-way plane tickets to create a roundtrip (often purchased from Expedia, Orbitz, etc.)
Travel Maximum Contributions
  • New England: $350
  • Other contiguous US: $850
  • Asia & Oceania: $2,000
  • All other locations: $1,500
  • Note these rates were updated in Spring 2023 and superseded any prior invitation rates. Any invitations without travel support will still not receive travel support.
Reimbursement Requests

Request Reimbursement with Cube

Refer to the back of your ID badge for more information. Checklists are available at the front desk and in the Reimbursement section of Cube.

Reimbursement Tips
  • Scanned original receipts are required for all expenses
  • Airfare receipt must show full itinerary and payment
  • ICERM does not offer per diem or meal reimbursement
  • Allowable mileage is reimbursed at prevailing IRS Business Rate and trip documented via pdf of Google Maps result
  • Keep all documentation until you receive your reimbursement!
Reimbursement Timing

6 - 8 weeks after all documentation is sent to ICERM. All reimbursement requests are reviewed by numerous central offices at Brown who may request additional documentation.

Reimbursement Deadline

Submissions must be received within 30 days of ICERM departure to avoid applicable taxes. Submissions after thirty days will incur applicable taxes. No submissions are accepted more than six months after the program end.