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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.02535 |
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| _version_ | 1866910814214029312 |
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| author | Lotnikov, Alexey Kotova, Anna |
| author_facet | Lotnikov, Alexey Kotova, Anna |
| contents | The article considers the Derrida-Retaux model with a random number of terms, i.e. a sequence of integer random variables defined by the relations $ X_{n + 1} = (X_n^{(1)} + X_n^{(2)} + ... + X_n^{(N_n)} - a)^{+}$, $n\ge 0$, where $X_n^{j}$ are independent copies of $X_n$, the values of $N_j$ are independent and identically distributed, $a$ is a positive integer. The energy in the model is defined as $Q:=\lim\limits_{n\to\infty} \frac{\mathbb{E}(X_{n})}{(\mathbb{E}N_1)^{n}}$. We present sufficient conditions (in terms of distributions of $X_0$ and $N_1$) for subcritical ($Q=0$) and supercritical ($Q>0$) regimes of model behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Criticality conditions in the Derrida-Retaux model with a random number of terms Lotnikov, Alexey Kotova, Anna Probability 60G50, 82B20, 82B27 The article considers the Derrida-Retaux model with a random number of terms, i.e. a sequence of integer random variables defined by the relations $ X_{n + 1} = (X_n^{(1)} + X_n^{(2)} + ... + X_n^{(N_n)} - a)^{+}$, $n\ge 0$, where $X_n^{j}$ are independent copies of $X_n$, the values of $N_j$ are independent and identically distributed, $a$ is a positive integer. The energy in the model is defined as $Q:=\lim\limits_{n\to\infty} \frac{\mathbb{E}(X_{n})}{(\mathbb{E}N_1)^{n}}$. We present sufficient conditions (in terms of distributions of $X_0$ and $N_1$) for subcritical ($Q=0$) and supercritical ($Q>0$) regimes of model behavior. |
| title | Criticality conditions in the Derrida-Retaux model with a random number of terms |
| topic | Probability 60G50, 82B20, 82B27 |
| url | https://arxiv.org/abs/2502.02535 |