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1. Painlevé-III Monodromy Maps Under the $$D_6\to D_8$$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

   https://doi.org/10.3842/SIGMA.2024.019  

   Barhoumi, Ahmad; Lisovyy, Oleg; Miller, Peter D; Prokhorov, Andrei (March 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   The third Painlevé equation in its generic form, often referred to as Painlevé-III($$D_6$$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $$u_0(x)$$ corresponding to parameters $$\alpha$$, $$\beta$$, denoted as the triple $$(u_0(x),\alpha,\beta)$$, we apply an explicit Bäcklund transformation to generate a family of solutions $$(u_n(x),\alpha + 4n,\beta + 4n)$$ indexed by $$n \in \mathbb N$$. We study the large $$n$$ behavior of the solutions $$(u_n(x), \alpha + 4n, \beta + 4n)$$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $$u_n(z/n)$$. Our main result is a proof that the limit of solutions $$u_n(z/n)$$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($$D_8$$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($$D_6$$), which are constructed using the seed solution $(1,4m,-4m)$ where $$m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $$D_6$$ and $$D_8$$ at $z = 0$. We also deduce the large $$n$$ behavior of the Umemura polynomials in a neighborhood of $z = 0$. 

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2. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

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3. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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4. Asymptotic Simplification of Aggregation-Diffusion Equations Towards the Heat kernel

   https://doi.org/10.1007/s00205-022-01838-5  

   Carrillo, José A.; Gómez-Castro, David; Yao, Yao; Zeng, Chongchun (January 2023, Archive for Rational Mechanics and Analysis)

   Abstract We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential$$W(x) \sim \log |x|$$ $W\left(x\right)\sim log\left|x\right|$for$$|x| \gg 1$$ $\left|x\right|\gg 1$appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the$$C^\alpha $$ $C^{\alpha }$regularity with a novel approach for this family of equations using modulus of continuity techniques. 

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5. Scaling limits and universality: Critical percolation on weighted graphs converging to an 𝐿³ graphon

   https://doi.org/10.1090/tran/9267  

   Baslingker, Jnaneshwar; Bhamidi, Shankar; Broutin, Nicolas; Sen, Sanchayan; Wang, Xuan (February 2025, Transactions of the American Mathematical Society)

   We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdős-Rényi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$graphon. In particular, we define a notion of the critical window in this setting. The $L^3$assumption ensures that the model is in the Erdős-Rényi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel’s RGIV model inside the critical window (see D.J. Aldous and B. Pittel [Random Structures Algorithms 17 (2000), pp. 79–102]). Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs (see E. Lubetzky and Y. Peled [Israel J. Math. 252 (2022), pp. 187–214]). In Bhamidi et al. [Scaling limits and universality II: geometry of maximal components in dynamic random graph models in the critical regime, In preparation], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Ruciński and Wormald’s random graph process with degree restrictions provided an additional technical condition about the barely subcritical behavior of this model can be proved (see A. Ruciński and N. C. Wormald [Combin. Probab. Comput. 1 (1992), pp. 169–180]). 

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