Follow-up on Hugo Duminil-Copin's Lecture: Probing SAW Universality and 4D Triviality

Overview

Following the suggestion from Professor Hugo Duminil-Copin, I sent a detailed email to him to follow up on the questions I was unable to ask during the Q&A session. The email, sent on November 27, 2025, covers two major topics: the subtle mathematical reasons behind the solvability of Self-Avoiding Walks (SAW) on the honeycomb lattice versus the square lattice (the Symmetry-Complexity Paradox), and a more conceptual inquiry into his work on the Marginal Triviality of the 4D $\Phi^4$ model in Quantum Field Theory. Since the email has not been answered, I plan to resolve these questions through self-study, and the formatted email below serves as the structured outline for that research.

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Detailed Summary of the Follow-Up Email

The email is structured into an introduction and two major sections, each containing a set of focused questions.

1. Introduction and Context

I identified myself as the undergraduate student from Nanjing University (NJU) who specifically traveled to Southeast University (SEU) for the talk and was the first to ask for a joint photograph. The purpose of the email was to convey my enthusiasm and follow up on the promised questions due to Professor Duminil-Copin's tight schedule.

2. Part I: SAW Universality and Lattice Geometry

This section dives deep into the Self-Avoiding Walk (SAW) problem, focusing on the contrast between the solvable honeycomb lattice and the unsolved square lattice.

• Core Fascination: The "analogy" thinking that led to the solution via the parafermionic operator was highlighted as the central point of interest.

• The Debate: I presented a summary of a debate with a roommate (Yuyang Su) regarding the underlying reason for the honeycomb lattice's solvability.

  • Opposing Argument: The roommate suggested solvability is merely due to the lower coordination number (3 choices on honeycomb versus 4 on square).
  • My Counter-Argument (Hypothesis): I argued that this simplistic view violates the Universality Principle of statistical mechanics and that historical examples (like the solvable Ising model on the square lattice) contradict the idea that local degrees of freedom are the primary barrier. I hypothesized that the solution hinges on a specific, non-obvious algebraic cancellation allowed by the honeycomb's geometry but absent in the square lattice.

• The Questions:

  1. Symmetry-Complexity Paradox: Why does the superior four-fold rotational symmetry of the square lattice appear to complicate the analysis, while the less symmetric honeycomb lattice permits a closed-form solution? Is there a deeper mathematical principle governing this inverse relationship?
  2. Methodological Extension: Since the current method does not apply to the square lattice, could a modified approach involving parafermionic operators, perhaps integrated with additional symmetries or disorder operators, be the key to future progress?

3. Part II: 4D Triviality and Scaling Limits

The second part transitions to Professor Duminil-Copin's work on fundamental physics, specifically the paper Aizenman, Duminil-Copin, Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models, which was recommended by members of the Chinese academic forum "Chaoli." I noted the connection to a forum member, Chengyang Shao, a postdoc at HDC's home institution, IHÉS.

• The Concept: The focus is on marginal triviality—the finding that the 4D $\Phi^4$ model behaves like a Gaussian (free) field in the scaling limit.

• The Questions:

  1. Geometric Intuition: From an intuitive or geometric standpoint (e.g., in the language of random currents or polymers), what specific feature in four dimensions causes the non-Gaussian interactions to effectively vanish, making 4D intersections appear "transparent" in the limit?
  2. SAW Connection: Given that the upper critical dimension for SAW is 4, does the $\Phi^4$ model's triviality imply that the "self-avoiding" constraint becomes geometrically negligible in 4D, making the polymer path behave essentially like a simple random walk?
  3. Dynamical Perspective (Critical Regularity Structures): A question posed by a forum member (Hachiri Tomoko) on the dynamic $\Phi^4$ equation: Considering the static triviality and the success of Martin Hairer’s Regularity Structures theory in sub-critical cases (like $\Phi^4_3$), is there a "critical version" of regularity structures (or an equivalent tool) that can rigorously describe the 4D dynamic $\Phi^4$ equation?

4. Conclusion

The email concluded with a polite acknowledgment of the volume of questions, stating that a brief comment or pointer to relevant literature would be sufficient, and reiterating my sincere enthusiasm.

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The Follow-up Email: Research Outline

Since the email has not yet been answered, it will now serve as my personal research outline for investigating these advanced topics.

Subject: Follow-up on your lecture at SEU: SAW Universality & 4D Triviality (Student from NJU)

From: Runnel Zhang runnel.zhang@smail.nju.edu.cn

Date: Thursday, November 27, 2025, 10:08 PM

To: Hugo Duminil-Copin hugo.duminil@unige.ch

Dear Professor Duminil-Copin,

I hope this email finds you well.

I am the undergraduate student from Nanjing University who attended your lecture at Southeast University on November 26th. I traveled from NJU specifically to hear your talk, and I was the one who rushed to the stage first to ask for a photo with you. It was truly an inspiring experience for me.

At that time, I had a few questions, but due to your tight schedule, you kindly suggested that I reach out via email. I am writing to follow up on that conversation.

My first set of questions concerns the honeycomb lattice Self-Avoiding Walks (SAW) part of your talk. It is such a legendary work, and I was fascinated by the "analogy" thinking using the parafermionic operator. While my background is not yet deep enough to fully grasp the rigorous proof, I had an interesting debate with my roommate regarding the intuition behind it.

My roommate Yuyang Su argued that the complexity of the SAW problem is primarily determined by the coordination number (the number of available directions at a vertex)—that the honeycomb lattice is solvable simply because a walker only has 3 choices, whereas other lattices have more. I respectfully disagreed with him. I argued that in statistical mechanics, difficulty rarely scales linearly with the number of local degrees of freedom. If it were just about counting paths, the principle of Universality would be compromised. I pointed out that historically, some integrable models (like the Ising model) were solved on square lattices despite having a higher coordination number, implying that "having more directions" is not the fundamental barrier. I suspect the honeycomb lattice succeeds because its geometry allows for a specific algebraic cancellation that the square lattice forbids—but I couldn't explain to him why the square lattice, with its superior symmetry, fails to support this structure.

Driven by this discussion, I have formulated the following questions:

1. The square lattice possesses a higher four-fold rotational symmetry compared to the honeycomb lattice, which intuitively should simplify the problem. Yet in reality, this seems to make it more challenging. Is there a deeper mathematical principle explaining why increased symmetry can sometimes complicate rather than simplify the analysis?
2. As you noted, the method does not apply directly to the square lattice, where the connective constant is still unknown. Do you think modified versions of the parafermionic approach—perhaps incorporating additional symmetries or disorder operators—could eventually yield progress there?

The second part of my email relates to your other work. I am the youngest core member of a small academic forum here in China called "Chaoli". After sharing my excitement about your lecture, our members recommended that I read your work on 4D physics: Aizenman M., Duminil-Copin H., Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models.

Interestingly, the introductory article on our forum regarding this paper (https://chaoli.club/index.php/6841/) was authored by Chengyang Shao, who I believe is currently a postdoc at IHÉS, your home institution!

Since my background in quantum field theory is still developing, I have a more intuitive question regarding this work:

1. The result establishes the "marginal triviality" (Gaussian behavior) in 4D. Intuitively, what specific geometric feature of the "random currents" or polymer representation in 4D causes the non-Gaussian interactions to vanish? Is there a visual way to understand why 4D intersections become "transparent" in the scaling limit compared to 3D?
2. This brings me back to your lecture on SAW. In 2D, the self-avoiding constraint fundamentally changes the universality class. However, since the upper critical dimension for SAW is 4, does this mean that in 4D, the "self-avoiding" constraint becomes geometrically negligible? Is the triviality of the $\phi^4$ field essentially stating that the polymer representation behaves like a simple random walk because the paths rarely intersect in such high dimensions?
3. One of our members nicknamed Hachiri Tomoko raised a question regarding the dynamical perspective. Since the static $\phi^4_4$ is trivial, how should we understand this via Stochastic Quantization? Martin Hairer’s Regularity Structure theory has successfully handled the sub-critical cases (like $\Phi^4_3$). The Question: Is there a "critical version" of regularity structures (or similar tools) that can rigorously describe the 4D dynamic $\phi^4$ equation? (This might be slightly tangential... I'm just the porteur of this question)

Thank you so much for your time.

I realize this email contains quite a few questions. Please do not feel pressured to answer them in detail. If you are busy, a brief comment or a pointer to relevant reading references would already be incredibly helpful. I do not wish to disturb your busy schedule but simply wanted to convey my enthusiasm.

Sincerely yours,

Runnel Zhang | Undergraduate Student Nanjing University | Jianxiong Academy 22 Hankou Road, Gulou District, Nanjing 210093, China Phone: +86 198 5277 1690 runnel.zhang@smail.nju.edu.cn | nju.edu.cn